On 03/12/2024 09:00 PM, olcott wrote:
> On 3/12/2024 10:49 PM, Ross Finlayson wrote:
>> On 03/12/2024 08:23 PM, Ross Finlayson wrote:
>>> On 03/12/2024 07:52 PM, olcott wrote:
>>>> On 3/12/2024 9:28 PM, Richard Damon wrote:
>>>>> On 3/12/24 4:31 PM, olcott wrote:
>>>>>> On 3/12/2024 6:11 PM, Richard Damon wrote:
>>>>>>> On 3/12/24 3:53 PM, olcott wrote:
>>>>>>>> On 3/12/2024 5:30 PM, Richard Damon wrote:
>>>>>>>>> On 3/12/24 2:34 PM, olcott wrote:
>>>>>>>>>> On 3/12/2024 4:23 PM, Richard Damon wrote:
>>>>>>>>>>> On 3/12/24 1:11 PM, olcott wrote:
>>>>>>>>>>>> On 3/12/2024 2:40 PM, Richard Damon wrote:
>>>>>>>>>>>>> On 3/12/24 12:02 PM, olcott wrote:
>>>>>>>>>>>>>> On 3/12/2024 1:31 PM, immibis wrote:
>>>>>>>>>>>>>>> On 12/03/24 19:12, olcott wrote:
>>>>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions |
>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> There is some input TMD to every H such that
>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> And it can be a different TMD to each H.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> When we disallow decider/input pairs that are incorrect
>>>>>>>>>>>>>>>> questions where both YES and NO are the wrong answer
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Once we understand that either YES or NO is the right
>>>>>>>>>>>>>>> answer, the whole rebuttal is tossed out as invalid and
>>>>>>>>>>>>>>> incorrect.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩
>>>>>>>>>>>>>> halts
>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn // Ĥ applied to ⟨Ĥ⟩ does
>>>>>>>>>>>>>> not halt
>>>>>>>>>>>>>> BOTH YES AND NO ARE THE WRONG ANSWER FOR EVERY Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩
>>>>>>>>>>>>>
>>>>>>>>>>>>> No, because a given H will only go to one of the answers. THAT
>>>>>>>>>>>>> will be wrong, and the other one right.
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions |
>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>
>>>>>>>>>>>> Not exactly. A pair of otherwise identical machines that
>>>>>>>>>>>> (that are contained within the above specified set)
>>>>>>>>>>>> only differ by return value will both be wrong on the
>>>>>>>>>>>> same pathological input.
>>>>>>>>>>>
>>>>>>>>>>> You mean a pair of DIFFERENT machines. Any difference is
>>>>>>>>>>> different.
>>>>>>>>>>
>>>>>>>>>> Every decider/input pair (referenced in the above set) has a
>>>>>>>>>> corresponding decider/input pair that only differs by the return
>>>>>>>>>> value of its decider.
>>>>>>>>>
>>>>>>>>> Nope.
>>>>>>>>>
>>>>>>>> ∀ H ∈ Turing_Machines_Returning_Boolean
>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions |
>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>
>>>>>>>> Every H/TMD pair (referenced in the above set) has a
>>>>>>>> corresponding H/TMD pair that only differs by the return
>>>>>>>> value of its Boolean_TM.
>>>>>>>
>>>>>>> That isn't in the set above.
>>>>>>>
>>>>>>>>
>>>>>>>> That both of these H/TMD pairs get the wrong answer proves that
>>>>>>>> their question was incorrect because the opposite answer to the
>>>>>>>> same question is also proven to be incorrect.
>>>>>>>>
>>>>>>>>
>>>>>>> Nope, since both aren't in the set selected.
>>>>>>>
>>>>>>
>>>>>> When they are deciders that must get the correct answer both
>>>>>> of them are not in the set.
>>>>>
>>>>> *IF* they are correct decider.
>>>>>
>>>>> WHen we select from all Turing Machine Deciders, there is no
>>>>> requirement that any of them get any particular answer right.
>>>>>
>>>>> So, ALL deciders are in the set that we cycle through and apply the
>>>>> following logic to ALL of them.
>>>>>
>>>>> Each is them paired with an input that it will get wrong, and the
>>>>> existance of the input was what as just proven, the ^ template
>>>>>
>>>>>>
>>>>>> When they are Turing_Machines_Returning_Boolean the this
>>>>>> set inherently includes identical pairs that only differ
>>>>>> by return value.
>>>>>
>>>>> But in the step of select and input that they will get wrong, they
>>>>> will be givne DIFFERENT inputs.
>>>>>
>>>>>>
>>>>>>> You just don't understand what that statement is saying.
>>>>>>>
>>>>>>> I've expalined it, but it seems over you head.
>>>>>>>
>>>>>> No the problem is that you are not paying attention.
>>>>>
>>>>> No, you keep on making STUPID mistakes, like thinking that select a
>>>>> input that the machine will get wrong needs to be the same for two
>>>>> differnt machines.
>>>>>
>>>>>
>>>>>
>>>>>>
>>>>>>> For Every H, we show we can find at least one input (chosen just for
>>>>>>> that machine) that it will get wrong.
>>>>>>>
>>>>>> When we use machine templates then we can see instances of
>>>>>> the same machine that only differs by return value where both
>>>>>> get the wrong answer on the same input. By same input I mean
>>>>>> the same finite string of numerical values.
>>>>>>
>>>>>
>>>>> But if they returned differnt values, they will have different
>>>>> descriptions.
>>>>>
>>>>> Otherwise, how could a UTM get the right answer, since it only gets
>>>>> the description.
>>>>
>>>> We can get around all of this stuff by simply using this criteria:
>>>> Date 10/13/2022 11:29:23 AM
>>>> *MIT Professor Michael Sipser agreed this verbatim paragraph is
>>>> correct*
>>>> (He has neither reviewed nor agreed to anything else in this paper)
>>>> (a) If simulating halt decider H correctly simulates its input D
>>>> until H
>>>> correctly determines that its simulated D would never stop running
>>>> unless aborted then
>>>> (b) H can abort its simulation of D and correctly report that D
>>>> specifies a non-halting sequence of configurations.
>>>>
>>>> *When we apply this criteria* (elaborated above)
>>>> Will you halt if you never abort your simulation?
>>>> *Then the halting problem is conquered*
>>>>
>>>> When two different machines implementing this criteria
>>>> get different results from identical inputs then we
>>>> know that Pathological Self-Reference has been detected.
>>>>
>>>> We don't even need to know that for:
>>>> *denial-of-service-attack detection*
>>>> *NO always means reject as unsafe*
>>>>
>>>
>>> But, Halting theorem never said "there's an input that halts
>>> all machines", it just says "for any machine, there's an input
>>> that halts it".
>>>
>>> Where "halt the machine" means "put it in an infinite loop".
>>>
>>> So, rather, Halting theorem never said "there's an input that
>>> exhausts all machines", it just says, "for any machine, there's
>>> an input that exhausts it".
>>>
>>> I still don't see how that would be with infinite tapes though,
>>> without a means of checking all the way right the tape in one
>>> step, i.e. that it's 1's or 0's or any pattern at all, any
>>> input that unbounded with respect to the machine basically
>>> exhausts it or where the machine would run in the unbounded.
>>>
>>>
>>> Of course any finite tape, has a static analysis that is
>>> not infinite, that decides whether or not it halts
>>> (or, loops, or grows, the state space of the decider).
>>>
>>> Static analysis has to either enumerate or _infer_ the
>>> state-space, where equal values in what's determined
>>> the idempotent can detect loops, while inequalities
>>> or proven divergence, ..., can detect unbounded growth.
>>>
>>> Now, proving convergence or divergence is its own kind
>>> of thing. For example, there are series that converge
>>> very slowly, and series that diverge very slowly. These
>>> are subtly intractable to analysis.
>>>
>>> Then the usual idea of progressions that rapidly grow
>>> yet return to what's detectable, are pathological to
>>> analysis, only in resources not correctness or completion,
>>> vis-a-vis the subtle intractability of the convergence or
>>> divergence what either halts, or loops, or grows unboundedly.
>>>
>>> Separately "not-halts" into "loops or grows unboundedly",
>>> has that really for most matters of interest of understanding
>>> the state-space of programs, is "halts or enters loops"
>>> and not "grows unboundedly".
>>>
>>> I.e. if the Halting problem is basically subject the
>>> subtle intractability of slow convergence, that otherwise
>>> it can just infer divergence and decide, practically
>>> it's sort of more relevant what the machine would be
>>> doing on the input on the tape, then with respect to
>>> beyond the Turing theory, of the state of the read-head,
>>> what happens when somebody modifies the tape, or events,
>>> the write-head.
>>>
>>> Anyways though for bounded inputs, besides slow divergence,
>>> it's to be made clear that _most all_ and _almost all_
>>> programs _are_ decided their behavior by static analysis.
>>>
>>> Though, "most all" and "almost all" might be a bit strong,
>>> but pretty much all that don't involve "the subtle intractability
>>> of slow divergence".
>>>
>>>
>>>
>>> Giving the idea that an existence result
>>> is in any way the expected result here
>>> seems sort of the root of this dilem-na.
>>>
>>>
>>>
>>>
>>>
>>>
>>> (Though that the real numbers in ZFC have a well-ordering
>>> and if they had a normal ordering that was a well-ordering,
>>> that would be a thing, because ZFC has a well-ordering of
>>> [0,1], but can't give one.)
>>>
>>>
>>
>>
>>
>> What I'm saying is that you'd need to solve all
>> cases of slow convergence and slow divergence
>> to have a correct and complete halt decider,
>> for the unbounded, and finite, if not the infinite.
>>
>> Most though setups of convergence though
>> would effectively exhaust according to the
>> existence of criteria of convergence as modeling
>> the computation, though, so, if you exclude
>> slow convergence and slow divergence,
>> then a subset of very usual machines is of
>> course quite all decidable. (Or decide-able,
>> or as I put it once, "not deciddable".)
>>
>>
>>
>> Well-ordering the reals in ZFC, that's it own
>> sort issue - that a well-ordering implies the
>> existence of a bijection from ordinals to a
>> set, then as that as tuples, subsets of those
>> are well-ordered by their ordinal part, then
>> that if ever uncountably many of those are
>> in normal order their real part and irrationals,
>> then between those are rationals. I.e.,
>> ZFC doesn't give an example of well-ordering
>> the reals, but there is one.
>>
>>
>> So, Peter, I think you're kind of misreading that
>> quote you authoritated. It just says "if static
>> analysis detects a loop or unboundedly growing
>> then it's not-halts".
>>
>>
>> Of course, for any bounded resource, there
>> are arbitrarily larger bounded resources
>> required by what's called pathological,
>> that an arbitrarily larger resource can though solve.
>>
>>
>>
>> So anyways "slow convergence" and "slow divergence",
>> have that in mathematics, it's unknown for some series
>> whether they converge or diverge, though it's usually
>> accepted that the inverse powers of 2 converge and
>> that the sum of integers diverges, and that periodic
>> functions do neither.
>>
>> I.e. the criteria of convergence and existence of a limit,
>> and the special case "diverges as going to infinity",
>> are not yet closed in today's mathematics.
>>
>> It's sort of a conjecture or independent whether
>> they ever could be, closed, and complete, the criteria.
>>
>>
>
> I have spent 20 years on The conventional halting problem proofs,
> Gödel 1931 Incompleteness and the Liar Paradox. I have only spent
> about seven years on Tarski Undefinability.
>
> The Halting problem is the only one of these that has a formal system
> that cannot have hidden gaps its its reasoning. If a machine can do it
> then it can be done.
>

On 3/13/2024 1:16 PM, Ross Finlayson wrote:
> On 03/12/2024 09:00 PM, olcott wrote:
>> On 3/12/2024 10:49 PM, Ross Finlayson wrote:
>>> On 03/12/2024 08:23 PM, Ross Finlayson wrote:
>>>> On 03/12/2024 07:52 PM, olcott wrote:
>>>>> On 3/12/2024 9:28 PM, Richard Damon wrote:
>>>>>> On 3/12/24 4:31 PM, olcott wrote:
>>>>>>> On 3/12/2024 6:11 PM, Richard Damon wrote:
>>>>>>>> On 3/12/24 3:53 PM, olcott wrote:
>>>>>>>>> On 3/12/2024 5:30 PM, Richard Damon wrote:
>>>>>>>>>> On 3/12/24 2:34 PM, olcott wrote:
>>>>>>>>>>> On 3/12/2024 4:23 PM, Richard Damon wrote:
>>>>>>>>>>>> On 3/12/24 1:11 PM, olcott wrote:
>>>>>>>>>>>>> On 3/12/2024 2:40 PM, Richard Damon wrote:
>>>>>>>>>>>>>> On 3/12/24 12:02 PM, olcott wrote:
>>>>>>>>>>>>>>> On 3/12/2024 1:31 PM, immibis wrote:
>>>>>>>>>>>>>>>> On 12/03/24 19:12, olcott wrote:
>>>>>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> There is some input TMD to every H such that
>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> And it can be a different TMD to each H.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> When we disallow decider/input pairs that are incorrect
>>>>>>>>>>>>>>>>> questions where both YES and NO are the wrong answer
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Once we understand that either YES or NO is the right
>>>>>>>>>>>>>>>> answer, the whole rebuttal is tossed out as invalid and
>>>>>>>>>>>>>>>> incorrect.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩
>>>>>>>>>>>>>>> halts
>>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn   // Ĥ applied to ⟨Ĥ⟩
>>>>>>>>>>>>>>> does
>>>>>>>>>>>>>>> not halt
>>>>>>>>>>>>>>> BOTH YES AND NO ARE THE WRONG ANSWER FOR EVERY Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> No, because a given H will only go to one of the answers.
>>>>>>>>>>>>>> THAT
>>>>>>>>>>>>>> will be wrong, and the other one right.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>
>>>>>>>>>>>>> Not exactly. A pair of otherwise identical machines that
>>>>>>>>>>>>> (that are contained within the above specified set)
>>>>>>>>>>>>> only differ by return value will both be wrong on the
>>>>>>>>>>>>> same pathological input.
>>>>>>>>>>>>
>>>>>>>>>>>> You mean a pair of DIFFERENT machines. Any difference is
>>>>>>>>>>>> different.
>>>>>>>>>>>
>>>>>>>>>>> Every decider/input pair (referenced in the above set) has a
>>>>>>>>>>> corresponding decider/input pair that only differs by the return
>>>>>>>>>>> value of its decider.
>>>>>>>>>>
>>>>>>>>>> Nope.
>>>>>>>>>>
>>>>>>>>> ∀ H ∈ Turing_Machines_Returning_Boolean
>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>
>>>>>>>>> Every H/TMD pair (referenced in the above set) has a
>>>>>>>>> corresponding H/TMD pair that only differs by the return
>>>>>>>>> value of its Boolean_TM.
>>>>>>>>
>>>>>>>> That isn't in the set above.
>>>>>>>>
>>>>>>>>>
>>>>>>>>> That both of these H/TMD pairs get the wrong answer proves that
>>>>>>>>> their question was incorrect because the opposite answer to the
>>>>>>>>> same question is also proven to be incorrect.
>>>>>>>>>
>>>>>>>>>
>>>>>>>> Nope, since both aren't in the set selected.
>>>>>>>>
>>>>>>>
>>>>>>> When they are deciders that must get the correct answer both
>>>>>>> of them are not in the set.
>>>>>>
>>>>>> *IF* they are correct decider.
>>>>>>
>>>>>> WHen we select from all Turing Machine Deciders, there is no
>>>>>> requirement that any of them get any particular answer right.
>>>>>>
>>>>>> So, ALL deciders are in the set that we cycle through and apply the
>>>>>> following logic to ALL of them.
>>>>>>
>>>>>> Each is them paired with an input that it will get wrong, and the
>>>>>> existance of the input was what as just proven, the ^ template
>>>>>>
>>>>>>>
>>>>>>> When they are Turing_Machines_Returning_Boolean the this
>>>>>>> set inherently includes identical pairs that only differ
>>>>>>> by return value.
>>>>>>
>>>>>> But in the step of select and input that they will get wrong, they
>>>>>> will be givne DIFFERENT inputs.
>>>>>>
>>>>>>>
>>>>>>>> You just don't understand what that statement is saying.
>>>>>>>>
>>>>>>>> I've expalined it, but it seems over you head.
>>>>>>>>
>>>>>>> No the problem is that you are not paying attention.
>>>>>>
>>>>>> No, you keep on making STUPID mistakes, like thinking that select a
>>>>>> input that the machine will get wrong needs to be the same for two
>>>>>> differnt machines.
>>>>>>
>>>>>>
>>>>>>
>>>>>>>
>>>>>>>> For Every H, we show we can find at least one input (chosen just
>>>>>>>> for
>>>>>>>> that machine) that it will get wrong.
>>>>>>>>
>>>>>>> When we use machine templates then we can see instances of
>>>>>>> the same machine that only differs by return value where both
>>>>>>> get the wrong answer on the same input. By same input I mean
>>>>>>> the same finite string of numerical values.
>>>>>>>
>>>>>>
>>>>>> But if they returned differnt values, they will have different
>>>>>> descriptions.
>>>>>>
>>>>>> Otherwise, how could a UTM get the right answer, since it only gets
>>>>>> the description.
>>>>>
>>>>> We can get around all of this stuff by simply using this criteria:
>>>>> Date 10/13/2022 11:29:23 AM
>>>>> *MIT Professor Michael Sipser agreed this verbatim paragraph is
>>>>> correct*
>>>>> (He has neither reviewed nor agreed to anything else in this paper)
>>>>> (a) If simulating halt decider H correctly simulates its input D
>>>>> until H
>>>>> correctly determines that its simulated D would never stop running
>>>>> unless aborted then
>>>>> (b) H can abort its simulation of D and correctly report that D
>>>>> specifies a non-halting sequence of configurations.
>>>>>
>>>>> *When we apply this criteria* (elaborated above)
>>>>> Will you halt if you never abort your simulation?
>>>>> *Then the halting problem is conquered*
>>>>>
>>>>> When two different machines implementing this criteria
>>>>> get different results from identical inputs then we
>>>>> know that Pathological Self-Reference has been detected.
>>>>>
>>>>> We don't even need to know that for:
>>>>> *denial-of-service-attack detection*
>>>>> *NO always means reject as unsafe*
>>>>>
>>>>
>>>> But, Halting theorem never said "there's an input that halts
>>>> all machines", it just says "for any machine, there's an input
>>>> that halts it".
>>>>
>>>> Where "halt the machine" means "put it in an infinite loop".
>>>>
>>>> So, rather, Halting theorem never said "there's an input that
>>>> exhausts all machines", it just says, "for any machine, there's
>>>> an input that exhausts it".
>>>>
>>>> I still don't see how that would be with infinite tapes though,
>>>> without a means of checking all the way right the tape in one
>>>> step, i.e. that it's 1's or 0's or any pattern at all, any
>>>> input that unbounded with respect to the machine basically
>>>> exhausts it or where the machine would run in the unbounded.
>>>>
>>>>
>>>> Of course any finite tape, has a static analysis that is
>>>> not infinite, that decides whether or not it halts
>>>> (or, loops, or grows, the state space of the decider).
>>>>
>>>> Static analysis has to either enumerate or _infer_ the
>>>> state-space, where equal values in what's determined
>>>> the idempotent can detect loops, while inequalities
>>>> or proven divergence, ..., can detect unbounded growth.
>>>>
>>>> Now, proving convergence or divergence is its own kind
>>>> of thing. For example, there are series that converge
>>>> very slowly, and series that diverge very slowly. These
>>>> are subtly intractable to analysis.
>>>>
>>>> Then the usual idea of progressions that rapidly grow
>>>> yet return to what's detectable, are pathological to
>>>> analysis, only in resources not correctness or completion,
>>>> vis-a-vis the subtle intractability of the convergence or
>>>> divergence what either halts, or loops, or grows unboundedly.
>>>>
>>>> Separately "not-halts" into "loops or grows unboundedly",
>>>> has that really for most matters of interest of understanding
>>>> the state-space of programs, is "halts or enters loops"
>>>> and not "grows unboundedly".
>>>>
>>>> I.e. if the Halting problem is basically subject the
>>>> subtle intractability of slow convergence, that otherwise
>>>> it can just infer divergence and decide, practically
>>>> it's sort of more relevant what the machine would be
>>>> doing on the input on the tape, then with respect to
>>>> beyond the Turing theory, of the state of the read-head,
>>>> what happens when somebody modifies the tape, or events,
>>>> the write-head.
>>>>
>>>> Anyways though for bounded inputs, besides slow divergence,
>>>> it's to be made clear that _most all_ and _almost all_
>>>> programs _are_ decided their behavior by static analysis.
>>>>
>>>> Though, "most all" and "almost all" might be a bit strong,
>>>> but pretty much all that don't involve "the subtle intractability
>>>> of slow divergence".
>>>>
>>>>
>>>>
>>>> Giving the idea that an existence result
>>>> is in any way the expected result here
>>>> seems sort of the root of this dilem-na.
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> (Though that the real numbers in ZFC have a well-ordering
>>>> and if they had a normal ordering that was a well-ordering,
>>>> that would be a thing, because ZFC has a well-ordering of
>>>> [0,1], but can't give one.)
>>>>
>>>>
>>>
>>>
>>>
>>> What I'm saying is that you'd need to solve all
>>> cases of slow convergence and slow divergence
>>> to have a correct and complete halt decider,
>>> for the unbounded, and finite, if not the infinite.
>>>
>>> Most though setups of convergence though
>>> would effectively exhaust according to the
>>> existence of criteria of convergence as modeling
>>> the computation, though, so, if you exclude
>>> slow convergence and slow divergence,
>>> then a subset of very usual machines is of
>>> course quite all decidable.  (Or decide-able,
>>> or as I put it once, "not deciddable".)
>>>
>>>
>>>
>>> Well-ordering the reals in ZFC, that's it own
>>> sort issue - that a well-ordering implies the
>>> existence of a bijection from ordinals to a
>>> set, then as that as tuples, subsets of those
>>> are well-ordered by their ordinal part, then
>>> that if ever uncountably many of those are
>>> in normal order their real part and irrationals,
>>> then between those are rationals.  I.e.,
>>> ZFC doesn't give an example of well-ordering
>>> the reals, but there is one.
>>>
>>>
>>> So, Peter, I think you're kind of misreading that
>>> quote you authoritated.  It just says "if static
>>> analysis detects a loop or unboundedly growing
>>> then it's not-halts".
>>>
>>>
>>> Of course, for any bounded resource, there
>>> are arbitrarily larger bounded resources
>>> required by what's called pathological,
>>> that an arbitrarily larger resource can though solve.
>>>
>>>
>>>
>>> So anyways "slow convergence" and "slow divergence",
>>> have that in mathematics, it's unknown for some series
>>> whether they converge or diverge, though it's usually
>>> accepted that the inverse powers of 2 converge and
>>> that the sum of integers diverges, and that periodic
>>> functions do neither.
>>>
>>> I.e. the criteria of convergence and existence of a limit,
>>> and the special case "diverges as going to infinity",
>>> are not yet closed in today's mathematics.
>>>
>>> It's sort of a conjecture or independent whether
>>> they ever could be, closed, and complete, the criteria.
>>>
>>>
>>
>> I have spent 20 years on The conventional halting problem proofs,
>> Gödel 1931 Incompleteness and the Liar Paradox. I have only spent
>> about seven years on Tarski Undefinability.
>>
>> The Halting problem is the only one of these that has a formal system
>> that cannot have hidden gaps its its reasoning. If a machine can do it
>> then it can be done.
>>
>
> "Algorithmics" is a study of algorithms. The most usual
> consideration is "asymptotics", "asymptotics of complexity",
> with the idea that computing does work, that there are
> closures and completions, and there are approximations and
> attenuations, as it were, in the full and the final and
> the initial and the partial.
>
> The most usual resources are time, and space, that actions
> occur in time, and according to a structure, in space.
>
> Of course any CS grad pretty much knows that.
>
>
> The objects of mathematics, have it so, that the finite
> and bounded resources of any collection of digital and
> binary machines, are finite and bounded, while, the unbounded
> models of mathematics that represent them, are unbounded,
> it's bad that anybody ever said a Turing tape was "infinite",
> except that it's unbounded as an object of mathematics, and
> the model of any finite, digital, binary machine.
>
> One might aver "digital" has a complement in computing,
> "analog", which represents the continuous instead of
> the discrete, and one might aver "binary" has a complement,
> either "real-valued" or "fuzzy" or variously about the
> "multi-valent", in logic, the models, according to the
> mechanisms, the models, their mechanics, machines.
>
> The Turing machine though is a model of unbounded machine
> as what any thusly simply mechanical model can implement,
> up to its bounds.
>
>
> So, the objects of mathematics, then involve analytical
> results. I can't draw a circle, though a compass may
> portray a diagram arbitrarily close, and no different.
> It's the analytical results, that make so for why
> mathematics is: "sensible, fungible, and tractable",
> and for "the ubiquitous success of mathematics" in
> all such matters of physics, which is a continuum mechanics,
> and as well discrete mechanics.
>
> Of course, results exist in mathematical objects,
> after the unbounded the unbounded and complete,
> which is why closures and completions are so
> relevant to all such matters, vis-a-vis approximations
> and attenuations, which in bounded terms result
> indistinguishably non-different results,
> up to precision, say, or tolerances.
>
> So, "machine" might be discrete, binary, and digital,
> or it might be continuous, multivalent, and analog.
>
> The Turing machine in a concrete form, is bounded.
>
> Similar models, here mostly for the stack models,
> stack machines, among finite-state-machines then
> for example concepts like "unbounded nondeterminism",
> here is pretty much about determinism, about the
> defined behavior of a machine according to a
> mathematical model that given defined behavior
> of the mechanisms results a mechanical model of
> a mathematical model, such a theory as this.
>
> So, the theory of computing, with mechanical or
> computational models, models of computing, is
> pretty much exactly like the theory of physics,
> with the attachment of physical models to the
> mathematical models, and rather, "interpreted as",
> the models, the models, with respect to each other.
>
> I.e. the extensionality that so follows "they're
> equal, or same, or not different, they represent
> each other, they're equi-interpretable, they model
> each other", the models, of logical and mathematical
> and computational and mechanical and physical models,
> help represent that over all the entire thing there
> is a usual theory for the entire thing, it's a theory
> where model theory models extensionality, and in identity
> intensionality, about equality, tautology, identity,
> and sameness/difference, and nearness/farness, and all
> these usual aspects of the mathematical models'
> arithmetic, algebra, and geometry. (And function
> theory and topology, with respect to categories and
> types, sets and parts, relations and predicates,
> then all the model-building among that which is
> all the propositional or the stipulated or axiomatic.)
>
> The idea is that this sort of platonistic universe of
> mathematical objects has always existed, and it's just
> discovered, then that axiomatics for example, just
> sort of results a model theory of it, with regards
> to models, the modular, modulation, modality, and the mode.
>
>
> So, a machine, with respect to computation, establishing
> the validity of interpretations of models, is subject to
> filling out all the above, vis-a-vis what it can do,
> and it can-do.
>
>
> Then, "slowly converging" and "slowly diverging"
> are examples that get get into that there are more
> law(s) of large numbers, than the usual classical
> law of large numbers.
>
> Some people variously do or don't have a mathematical
> model of larger numbers, or the infinite, at all.
> It's a totally ancient and dogmatic tradition that
> no, we finite peoples, persons, or agents of limited
> and finite means, no we do not have discrete, digital,
> binary mechanical models of the infinite.
>
> Yet, it's also about the first thing that deduction
> arrives at just from the plain contemplation of the
> beyond the unbounded, like the theories of Democritus
> and Atomism or the theories of Zeno and Continuity,
> that make it so we at least have something to arrive
> at for models of the unbounded, as "methods of exhaustion",
> the approximation and attenuative what results
> the asymptotics of algorithmics.
>
> So, we do have mental models of the continuous and infinite.
> And, where physics is a continuum mechanics,
> there are physical models of it as well, then
> with regards to the philosophy of science and
> the scientific method and the Big Science and
> from the Primary Science, lots to establish
> that the continuous and analog has mathematical
> results, as whethe, "sensible, fungible, and tractable",
> to machines at all.
>
> Numerical methods then, and, approximation and
> attenuation, result from analysis, why they exist,
> here for the criteria of convergence and existence
> of limits, in the closed, in the complete.
>
>
> So, a metonymy, unless it's "the Strong Metonymy",
> which is utter intensionality of "the ground model",
> is yet a metaphor and a metaphor yet a weaker metonymy
> joint among weaker metonymys, that, ontology, has
> that there are or may be a completion, of ontology,
> as a "strong ontology" and "strong metonymy",
> but otherwise it's just a bag of facts.
>
>
>
> Here then there's certainly a perceived analytical
> development Cantor space, and, Square Cantor space,
> and, Signal Cantor space as it were, about that among
> the law(s) of large numbers, there are definitions of
> continuity, at least including field-reals, line-reals,
> and signal-reals, three sorts continuous domains.
>
>
> Mathematics owes physics this to better begin to
> explain, to disclose, to find truths of, continuum mechanics.
>
>
> Then that analog, fuzzy, multi-value, "quantum" computing
> models (parts) of that, that discrete, binary, digital
> computing does not, is a thing.
>
>
>
> The convergence of terms in series is an example
> of a model of things, that, sometimes has clear
> and direct and perfectly accurate representations,
> models, in stack machine, and sometimes doesn't.
>
>
> Of course, for any bounded input, there is a sufficiently
> larger bounded analyzer, that does solve it. So, why not
> just put those all together? It would be infinite,
> yet most things are not. (Or, you know, are.)
>
>
> It's called "foundations" the study of all these things
> as a fundamental coherency, a thing together, while of
> its parts.
>
> So, students should well have the concepts of "abacus"
> and "slide rule" and other "tabulators and computers"
> of the usual manual mechanical variety down, according
> to the principles of the mathematical models behind them,
> then introducing the law(s) of large numbers, then
> introducing the idea of a standard model of an infinite,
> about rates, related rates, asymptotics, and bounds.
>
>
> "Foundations" then the idea is that it's been around
> forever, it's our dogma and from our canon and it develops
> over time, and the current edition is called "modern".

On 3/13/24 10:20 PM, olcott wrote:
> On 3/13/2024 1:16 PM, Ross Finlayson wrote:
>> On 03/12/2024 09:00 PM, olcott wrote:
>>> On 3/12/2024 10:49 PM, Ross Finlayson wrote:
>>>> On 03/12/2024 08:23 PM, Ross Finlayson wrote:
>>>>> On 03/12/2024 07:52 PM, olcott wrote:
>>>>>> On 3/12/2024 9:28 PM, Richard Damon wrote:
>>>>>>> On 3/12/24 4:31 PM, olcott wrote:
>>>>>>>> On 3/12/2024 6:11 PM, Richard Damon wrote:
>>>>>>>>> On 3/12/24 3:53 PM, olcott wrote:
>>>>>>>>>> On 3/12/2024 5:30 PM, Richard Damon wrote:
>>>>>>>>>>> On 3/12/24 2:34 PM, olcott wrote:
>>>>>>>>>>>> On 3/12/2024 4:23 PM, Richard Damon wrote:
>>>>>>>>>>>>> On 3/12/24 1:11 PM, olcott wrote:
>>>>>>>>>>>>>> On 3/12/2024 2:40 PM, Richard Damon wrote:
>>>>>>>>>>>>>>> On 3/12/24 12:02 PM, olcott wrote:
>>>>>>>>>>>>>>>> On 3/12/2024 1:31 PM, immibis wrote:
>>>>>>>>>>>>>>>>> On 12/03/24 19:12, olcott wrote:
>>>>>>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> There is some input TMD to every H such that
>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> And it can be a different TMD to each H.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> When we disallow decider/input pairs that are incorrect
>>>>>>>>>>>>>>>>>> questions where both YES and NO are the wrong answer
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Once we understand that either YES or NO is the right
>>>>>>>>>>>>>>>>> answer, the whole rebuttal is tossed out as invalid and
>>>>>>>>>>>>>>>>> incorrect.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩
>>>>>>>>>>>>>>>> halts
>>>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn   // Ĥ applied to ⟨Ĥ⟩
>>>>>>>>>>>>>>>> does
>>>>>>>>>>>>>>>> not halt
>>>>>>>>>>>>>>>> BOTH YES AND NO ARE THE WRONG ANSWER FOR EVERY Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> No, because a given H will only go to one of the answers.
>>>>>>>>>>>>>>> THAT
>>>>>>>>>>>>>>> will be wrong, and the other one right.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Not exactly. A pair of otherwise identical machines that
>>>>>>>>>>>>>> (that are contained within the above specified set)
>>>>>>>>>>>>>> only differ by return value will both be wrong on the
>>>>>>>>>>>>>> same pathological input.
>>>>>>>>>>>>>
>>>>>>>>>>>>> You mean a pair of DIFFERENT machines. Any difference is
>>>>>>>>>>>>> different.
>>>>>>>>>>>>
>>>>>>>>>>>> Every decider/input pair (referenced in the above set) has a
>>>>>>>>>>>> corresponding decider/input pair that only differs by the
>>>>>>>>>>>> return
>>>>>>>>>>>> value of its decider.
>>>>>>>>>>>
>>>>>>>>>>> Nope.
>>>>>>>>>>>
>>>>>>>>>> ∀ H ∈ Turing_Machines_Returning_Boolean
>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>
>>>>>>>>>> Every H/TMD pair (referenced in the above set) has a
>>>>>>>>>> corresponding H/TMD pair that only differs by the return
>>>>>>>>>> value of its Boolean_TM.
>>>>>>>>>
>>>>>>>>> That isn't in the set above.
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> That both of these H/TMD pairs get the wrong answer proves that
>>>>>>>>>> their question was incorrect because the opposite answer to the
>>>>>>>>>> same question is also proven to be incorrect.
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>> Nope, since both aren't in the set selected.
>>>>>>>>>
>>>>>>>>
>>>>>>>> When they are deciders that must get the correct answer both
>>>>>>>> of them are not in the set.
>>>>>>>
>>>>>>> *IF* they are correct decider.
>>>>>>>
>>>>>>> WHen we select from all Turing Machine Deciders, there is no
>>>>>>> requirement that any of them get any particular answer right.
>>>>>>>
>>>>>>> So, ALL deciders are in the set that we cycle through and apply the
>>>>>>> following logic to ALL of them.
>>>>>>>
>>>>>>> Each is them paired with an input that it will get wrong, and the
>>>>>>> existance of the input was what as just proven, the ^ template
>>>>>>>
>>>>>>>>
>>>>>>>> When they are Turing_Machines_Returning_Boolean the this
>>>>>>>> set inherently includes identical pairs that only differ
>>>>>>>> by return value.
>>>>>>>
>>>>>>> But in the step of select and input that they will get wrong, they
>>>>>>> will be givne DIFFERENT inputs.
>>>>>>>
>>>>>>>>
>>>>>>>>> You just don't understand what that statement is saying.
>>>>>>>>>
>>>>>>>>> I've expalined it, but it seems over you head.
>>>>>>>>>
>>>>>>>> No the problem is that you are not paying attention.
>>>>>>>
>>>>>>> No, you keep on making STUPID mistakes, like thinking that select a
>>>>>>> input that the machine will get wrong needs to be the same for two
>>>>>>> differnt machines.
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>>
>>>>>>>>> For Every H, we show we can find at least one input (chosen
>>>>>>>>> just for
>>>>>>>>> that machine) that it will get wrong.
>>>>>>>>>
>>>>>>>> When we use machine templates then we can see instances of
>>>>>>>> the same machine that only differs by return value where both
>>>>>>>> get the wrong answer on the same input. By same input I mean
>>>>>>>> the same finite string of numerical values.
>>>>>>>>
>>>>>>>
>>>>>>> But if they returned differnt values, they will have different
>>>>>>> descriptions.
>>>>>>>
>>>>>>> Otherwise, how could a UTM get the right answer, since it only gets
>>>>>>> the description.
>>>>>>
>>>>>> We can get around all of this stuff by simply using this criteria:
>>>>>> Date 10/13/2022 11:29:23 AM
>>>>>> *MIT Professor Michael Sipser agreed this verbatim paragraph is
>>>>>> correct*
>>>>>> (He has neither reviewed nor agreed to anything else in this paper)
>>>>>> (a) If simulating halt decider H correctly simulates its input D
>>>>>> until H
>>>>>> correctly determines that its simulated D would never stop running
>>>>>> unless aborted then
>>>>>> (b) H can abort its simulation of D and correctly report that D
>>>>>> specifies a non-halting sequence of configurations.
>>>>>>
>>>>>> *When we apply this criteria* (elaborated above)
>>>>>> Will you halt if you never abort your simulation?
>>>>>> *Then the halting problem is conquered*
>>>>>>
>>>>>> When two different machines implementing this criteria
>>>>>> get different results from identical inputs then we
>>>>>> know that Pathological Self-Reference has been detected.
>>>>>>
>>>>>> We don't even need to know that for:
>>>>>> *denial-of-service-attack detection*
>>>>>> *NO always means reject as unsafe*
>>>>>>
>>>>>
>>>>> But, Halting theorem never said "there's an input that halts
>>>>> all machines", it just says "for any machine, there's an input
>>>>> that halts it".
>>>>>
>>>>> Where "halt the machine" means "put it in an infinite loop".
>>>>>
>>>>> So, rather, Halting theorem never said "there's an input that
>>>>> exhausts all machines", it just says, "for any machine, there's
>>>>> an input that exhausts it".
>>>>>
>>>>> I still don't see how that would be with infinite tapes though,
>>>>> without a means of checking all the way right the tape in one
>>>>> step, i.e. that it's 1's or 0's or any pattern at all, any
>>>>> input that unbounded with respect to the machine basically
>>>>> exhausts it or where the machine would run in the unbounded.
>>>>>
>>>>>
>>>>> Of course any finite tape, has a static analysis that is
>>>>> not infinite, that decides whether or not it halts
>>>>> (or, loops, or grows, the state space of the decider).
>>>>>
>>>>> Static analysis has to either enumerate or _infer_ the
>>>>> state-space, where equal values in what's determined
>>>>> the idempotent can detect loops, while inequalities
>>>>> or proven divergence, ..., can detect unbounded growth.
>>>>>
>>>>> Now, proving convergence or divergence is its own kind
>>>>> of thing. For example, there are series that converge
>>>>> very slowly, and series that diverge very slowly. These
>>>>> are subtly intractable to analysis.
>>>>>
>>>>> Then the usual idea of progressions that rapidly grow
>>>>> yet return to what's detectable, are pathological to
>>>>> analysis, only in resources not correctness or completion,
>>>>> vis-a-vis the subtle intractability of the convergence or
>>>>> divergence what either halts, or loops, or grows unboundedly.
>>>>>
>>>>> Separately "not-halts" into "loops or grows unboundedly",
>>>>> has that really for most matters of interest of understanding
>>>>> the state-space of programs, is "halts or enters loops"
>>>>> and not "grows unboundedly".
>>>>>
>>>>> I.e. if the Halting problem is basically subject the
>>>>> subtle intractability of slow convergence, that otherwise
>>>>> it can just infer divergence and decide, practically
>>>>> it's sort of more relevant what the machine would be
>>>>> doing on the input on the tape, then with respect to
>>>>> beyond the Turing theory, of the state of the read-head,
>>>>> what happens when somebody modifies the tape, or events,
>>>>> the write-head.
>>>>>
>>>>> Anyways though for bounded inputs, besides slow divergence,
>>>>> it's to be made clear that _most all_ and _almost all_
>>>>> programs _are_ decided their behavior by static analysis.
>>>>>
>>>>> Though, "most all" and "almost all" might be a bit strong,
>>>>> but pretty much all that don't involve "the subtle intractability
>>>>> of slow divergence".
>>>>>
>>>>>
>>>>>
>>>>> Giving the idea that an existence result
>>>>> is in any way the expected result here
>>>>> seems sort of the root of this dilem-na.
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> (Though that the real numbers in ZFC have a well-ordering
>>>>> and if they had a normal ordering that was a well-ordering,
>>>>> that would be a thing, because ZFC has a well-ordering of
>>>>> [0,1], but can't give one.)
>>>>>
>>>>>
>>>>
>>>>
>>>>
>>>> What I'm saying is that you'd need to solve all
>>>> cases of slow convergence and slow divergence
>>>> to have a correct and complete halt decider,
>>>> for the unbounded, and finite, if not the infinite.
>>>>
>>>> Most though setups of convergence though
>>>> would effectively exhaust according to the
>>>> existence of criteria of convergence as modeling
>>>> the computation, though, so, if you exclude
>>>> slow convergence and slow divergence,
>>>> then a subset of very usual machines is of
>>>> course quite all decidable.  (Or decide-able,
>>>> or as I put it once, "not deciddable".)
>>>>
>>>>
>>>>
>>>> Well-ordering the reals in ZFC, that's it own
>>>> sort issue - that a well-ordering implies the
>>>> existence of a bijection from ordinals to a
>>>> set, then as that as tuples, subsets of those
>>>> are well-ordered by their ordinal part, then
>>>> that if ever uncountably many of those are
>>>> in normal order their real part and irrationals,
>>>> then between those are rationals.  I.e.,
>>>> ZFC doesn't give an example of well-ordering
>>>> the reals, but there is one.
>>>>
>>>>
>>>> So, Peter, I think you're kind of misreading that
>>>> quote you authoritated.  It just says "if static
>>>> analysis detects a loop or unboundedly growing
>>>> then it's not-halts".
>>>>
>>>>
>>>> Of course, for any bounded resource, there
>>>> are arbitrarily larger bounded resources
>>>> required by what's called pathological,
>>>> that an arbitrarily larger resource can though solve.
>>>>
>>>>
>>>>
>>>> So anyways "slow convergence" and "slow divergence",
>>>> have that in mathematics, it's unknown for some series
>>>> whether they converge or diverge, though it's usually
>>>> accepted that the inverse powers of 2 converge and
>>>> that the sum of integers diverges, and that periodic
>>>> functions do neither.
>>>>
>>>> I.e. the criteria of convergence and existence of a limit,
>>>> and the special case "diverges as going to infinity",
>>>> are not yet closed in today's mathematics.
>>>>
>>>> It's sort of a conjecture or independent whether
>>>> they ever could be, closed, and complete, the criteria.
>>>>
>>>>
>>>
>>> I have spent 20 years on The conventional halting problem proofs,
>>> Gödel 1931 Incompleteness and the Liar Paradox. I have only spent
>>> about seven years on Tarski Undefinability.
>>>
>>> The Halting problem is the only one of these that has a formal system
>>> that cannot have hidden gaps its its reasoning. If a machine can do it
>>> then it can be done.
>>>
>>
>> "Algorithmics" is a study of algorithms. The most usual
>> consideration is "asymptotics", "asymptotics of complexity",
>> with the idea that computing does work, that there are
>> closures and completions, and there are approximations and
>> attenuations, as it were, in the full and the final and
>> the initial and the partial.
>>
>> The most usual resources are time, and space, that actions
>> occur in time, and according to a structure, in space.
>>
>> Of course any CS grad pretty much knows that.
>>
>>
>> The objects of mathematics, have it so, that the finite
>> and bounded resources of any collection of digital and
>> binary machines, are finite and bounded, while, the unbounded
>> models of mathematics that represent them, are unbounded,
>> it's bad that anybody ever said a Turing tape was "infinite",
>> except that it's unbounded as an object of mathematics, and
>> the model of any finite, digital, binary machine.
>>
>> One might aver "digital" has a complement in computing,
>> "analog", which represents the continuous instead of
>> the discrete, and one might aver "binary" has a complement,
>> either "real-valued" or "fuzzy" or variously about the
>> "multi-valent", in logic, the models, according to the
>> mechanisms, the models, their mechanics, machines.
>>
>> The Turing machine though is a model of unbounded machine
>> as what any thusly simply mechanical model can implement,
>> up to its bounds.
>>
>>
>> So, the objects of mathematics, then involve analytical
>> results. I can't draw a circle, though a compass may
>> portray a diagram arbitrarily close, and no different.
>> It's the analytical results, that make so for why
>> mathematics is: "sensible, fungible, and tractable",
>> and for "the ubiquitous success of mathematics" in
>> all such matters of physics, which is a continuum mechanics,
>> and as well discrete mechanics.
>>
>> Of course, results exist in mathematical objects,
>> after the unbounded the unbounded and complete,
>> which is why closures and completions are so
>> relevant to all such matters, vis-a-vis approximations
>> and attenuations, which in bounded terms result
>> indistinguishably non-different results,
>> up to precision, say, or tolerances.
>>
>> So, "machine" might be discrete, binary, and digital,
>> or it might be continuous, multivalent, and analog.
>>
>> The Turing machine in a concrete form, is bounded.
>>
>> Similar models, here mostly for the stack models,
>> stack machines, among finite-state-machines then
>> for example concepts like "unbounded nondeterminism",
>> here is pretty much about determinism, about the
>> defined behavior of a machine according to a
>> mathematical model that given defined behavior
>> of the mechanisms results a mechanical model of
>> a mathematical model, such a theory as this.
>>
>> So, the theory of computing, with mechanical or
>> computational models, models of computing, is
>> pretty much exactly like the theory of physics,
>> with the attachment of physical models to the
>> mathematical models, and rather, "interpreted as",
>> the models, the models, with respect to each other.
>>
>> I.e. the extensionality that so follows "they're
>> equal, or same, or not different, they represent
>> each other, they're equi-interpretable, they model
>> each other", the models, of logical and mathematical
>> and computational and mechanical and physical models,
>> help represent that over all the entire thing there
>> is a usual theory for the entire thing, it's a theory
>> where model theory models extensionality, and in identity
>> intensionality, about equality, tautology, identity,
>> and sameness/difference, and nearness/farness, and all
>> these usual aspects of the mathematical models'
>> arithmetic, algebra, and geometry. (And function
>> theory and topology, with respect to categories and
>> types, sets and parts, relations and predicates,
>> then all the model-building among that which is
>> all the propositional or the stipulated or axiomatic.)
>>
>> The idea is that this sort of platonistic universe of
>> mathematical objects has always existed, and it's just
>> discovered, then that axiomatics for example, just
>> sort of results a model theory of it, with regards
>> to models, the modular, modulation, modality, and the mode.
>>
>>
>> So, a machine, with respect to computation, establishing
>> the validity of interpretations of models, is subject to
>> filling out all the above, vis-a-vis what it can do,
>> and it can-do.
>>
>>
>> Then, "slowly converging" and "slowly diverging"
>> are examples that get get into that there are more
>> law(s) of large numbers, than the usual classical
>> law of large numbers.
>>
>> Some people variously do or don't have a mathematical
>> model of larger numbers, or the infinite, at all.
>> It's a totally ancient and dogmatic tradition that
>> no, we finite peoples, persons, or agents of limited
>> and finite means, no we do not have discrete, digital,
>> binary mechanical models of the infinite.
>>
>> Yet, it's also about the first thing that deduction
>> arrives at just from the plain contemplation of the
>> beyond the unbounded, like the theories of Democritus
>> and Atomism or the theories of Zeno and Continuity,
>> that make it so we at least have something to arrive
>> at for models of the unbounded, as "methods of exhaustion",
>> the approximation and attenuative what results
>> the asymptotics of algorithmics.
>>
>> So, we do have mental models of the continuous and infinite.
>> And, where physics is a continuum mechanics,
>> there are physical models of it as well, then
>> with regards to the philosophy of science and
>> the scientific method and the Big Science and
>> from the Primary Science, lots to establish
>> that the continuous and analog has mathematical
>> results, as whethe, "sensible, fungible, and tractable",
>> to machines at all.
>>
>> Numerical methods then, and, approximation and
>> attenuation, result from analysis, why they exist,
>> here for the criteria of convergence and existence
>> of limits, in the closed, in the complete.
>>
>>
>> So, a metonymy, unless it's "the Strong Metonymy",
>> which is utter intensionality of "the ground model",
>> is yet a metaphor and a metaphor yet a weaker metonymy
>> joint among weaker metonymys, that, ontology, has
>> that there are or may be a completion, of ontology,
>> as a "strong ontology" and "strong metonymy",
>> but otherwise it's just a bag of facts.
>>
>>
>>
>> Here then there's certainly a perceived analytical
>> development Cantor space, and, Square Cantor space,
>> and, Signal Cantor space as it were, about that among
>> the law(s) of large numbers, there are definitions of
>> continuity, at least including field-reals, line-reals,
>> and signal-reals, three sorts continuous domains.
>>
>>
>> Mathematics owes physics this to better begin to
>> explain, to disclose, to find truths of, continuum mechanics.
>>
>>
>> Then that analog, fuzzy, multi-value, "quantum" computing
>> models (parts) of that, that discrete, binary, digital
>> computing does not, is a thing.
>>
>>
>>
>> The convergence of terms in series is an example
>> of a model of things, that, sometimes has clear
>> and direct and perfectly accurate representations,
>> models, in stack machine, and sometimes doesn't.
>>
>>
>> Of course, for any bounded input, there is a sufficiently
>> larger bounded analyzer, that does solve it. So, why not
>> just put those all together? It would be infinite,
>> yet most things are not. (Or, you know, are.)
>>
>>
>> It's called "foundations" the study of all these things
>> as a fundamental coherency, a thing together, while of
>> its parts.
>>
>> So, students should well have the concepts of "abacus"
>> and "slide rule" and other "tabulators and computers"
>> of the usual manual mechanical variety down, according
>> to the principles of the mathematical models behind them,
>> then introducing the law(s) of large numbers, then
>> introducing the idea of a standard model of an infinite,
>> about rates, related rates, asymptotics, and bounds.
>>
>>
>> "Foundations" then the idea is that it's been around
>> forever, it's our dogma and from our canon and it develops
>> over time, and the current edition is called "modern".
>
> Did I make any mistakes?
>
> Tarski anchors his proof in the Liar Paradox
> https://liarparadox.org/Tarski_247_248.pdf
>
> How Tarski encodes the Liar Paradox
> x ∉ True if and only if p
> where the symbol 'p' represents the whole sentence x
>
> This is transformed into line 01 of his proof
> by replacing True with Provable
>
> *Here is the Tarski Undefinability Theorem proof*
> (1) x ∉ Provable if and only if p   // assumption
> (2) x ∈ True if and only if p       // assumption
> (3) x ∉ Provable if and only if x ∈ True.
> (4) either x ∉ True or x̄ ∉ True;    // axiom: ~True(x) ∨ ~True(~x)
> (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
> (6) if x̄ ∈ Provable, then x̄ ∈ True;  // axiom: Provable(~x) → True(~x)
> (7) x ∈ True
> (8) x ∉ Provable
> (9) x̄ ∉ Provable
>
> Tarski's Full proof
> https://liarparadox.org/Tarski_275_276.pdf
>
>
>

On 03/13/2024 10:20 PM, olcott wrote:
> On 3/13/2024 1:16 PM, Ross Finlayson wrote:
>> On 03/12/2024 09:00 PM, olcott wrote:
>>> On 3/12/2024 10:49 PM, Ross Finlayson wrote:
>>>> On 03/12/2024 08:23 PM, Ross Finlayson wrote:
>>>>> On 03/12/2024 07:52 PM, olcott wrote:
>>>>>> On 3/12/2024 9:28 PM, Richard Damon wrote:
>>>>>>> On 3/12/24 4:31 PM, olcott wrote:
>>>>>>>> On 3/12/2024 6:11 PM, Richard Damon wrote:
>>>>>>>>> On 3/12/24 3:53 PM, olcott wrote:
>>>>>>>>>> On 3/12/2024 5:30 PM, Richard Damon wrote:
>>>>>>>>>>> On 3/12/24 2:34 PM, olcott wrote:
>>>>>>>>>>>> On 3/12/2024 4:23 PM, Richard Damon wrote:
>>>>>>>>>>>>> On 3/12/24 1:11 PM, olcott wrote:
>>>>>>>>>>>>>> On 3/12/2024 2:40 PM, Richard Damon wrote:
>>>>>>>>>>>>>>> On 3/12/24 12:02 PM, olcott wrote:
>>>>>>>>>>>>>>>> On 3/12/2024 1:31 PM, immibis wrote:
>>>>>>>>>>>>>>>>> On 12/03/24 19:12, olcott wrote:
>>>>>>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions |
>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> There is some input TMD to every H such that
>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> And it can be a different TMD to each H.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> When we disallow decider/input pairs that are incorrect
>>>>>>>>>>>>>>>>>> questions where both YES and NO are the wrong answer
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Once we understand that either YES or NO is the right
>>>>>>>>>>>>>>>>> answer, the whole rebuttal is tossed out as invalid and
>>>>>>>>>>>>>>>>> incorrect.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩
>>>>>>>>>>>>>>>> halts
>>>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn // Ĥ applied to ⟨Ĥ⟩
>>>>>>>>>>>>>>>> does
>>>>>>>>>>>>>>>> not halt
>>>>>>>>>>>>>>>> BOTH YES AND NO ARE THE WRONG ANSWER FOR EVERY Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> No, because a given H will only go to one of the answers.
>>>>>>>>>>>>>>> THAT
>>>>>>>>>>>>>>> will be wrong, and the other one right.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions |
>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Not exactly. A pair of otherwise identical machines that
>>>>>>>>>>>>>> (that are contained within the above specified set)
>>>>>>>>>>>>>> only differ by return value will both be wrong on the
>>>>>>>>>>>>>> same pathological input.
>>>>>>>>>>>>>
>>>>>>>>>>>>> You mean a pair of DIFFERENT machines. Any difference is
>>>>>>>>>>>>> different.
>>>>>>>>>>>>
>>>>>>>>>>>> Every decider/input pair (referenced in the above set) has a
>>>>>>>>>>>> corresponding decider/input pair that only differs by the
>>>>>>>>>>>> return
>>>>>>>>>>>> value of its decider.
>>>>>>>>>>>
>>>>>>>>>>> Nope.
>>>>>>>>>>>
>>>>>>>>>> ∀ H ∈ Turing_Machines_Returning_Boolean
>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions |
>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>
>>>>>>>>>> Every H/TMD pair (referenced in the above set) has a
>>>>>>>>>> corresponding H/TMD pair that only differs by the return
>>>>>>>>>> value of its Boolean_TM.
>>>>>>>>>
>>>>>>>>> That isn't in the set above.
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> That both of these H/TMD pairs get the wrong answer proves that
>>>>>>>>>> their question was incorrect because the opposite answer to the
>>>>>>>>>> same question is also proven to be incorrect.
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>> Nope, since both aren't in the set selected.
>>>>>>>>>
>>>>>>>>
>>>>>>>> When they are deciders that must get the correct answer both
>>>>>>>> of them are not in the set.
>>>>>>>
>>>>>>> *IF* they are correct decider.
>>>>>>>
>>>>>>> WHen we select from all Turing Machine Deciders, there is no
>>>>>>> requirement that any of them get any particular answer right.
>>>>>>>
>>>>>>> So, ALL deciders are in the set that we cycle through and apply the
>>>>>>> following logic to ALL of them.
>>>>>>>
>>>>>>> Each is them paired with an input that it will get wrong, and the
>>>>>>> existance of the input was what as just proven, the ^ template
>>>>>>>
>>>>>>>>
>>>>>>>> When they are Turing_Machines_Returning_Boolean the this
>>>>>>>> set inherently includes identical pairs that only differ
>>>>>>>> by return value.
>>>>>>>
>>>>>>> But in the step of select and input that they will get wrong, they
>>>>>>> will be givne DIFFERENT inputs.
>>>>>>>
>>>>>>>>
>>>>>>>>> You just don't understand what that statement is saying.
>>>>>>>>>
>>>>>>>>> I've expalined it, but it seems over you head.
>>>>>>>>>
>>>>>>>> No the problem is that you are not paying attention.
>>>>>>>
>>>>>>> No, you keep on making STUPID mistakes, like thinking that select a
>>>>>>> input that the machine will get wrong needs to be the same for two
>>>>>>> differnt machines.
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>>
>>>>>>>>> For Every H, we show we can find at least one input (chosen
>>>>>>>>> just for
>>>>>>>>> that machine) that it will get wrong.
>>>>>>>>>
>>>>>>>> When we use machine templates then we can see instances of
>>>>>>>> the same machine that only differs by return value where both
>>>>>>>> get the wrong answer on the same input. By same input I mean
>>>>>>>> the same finite string of numerical values.
>>>>>>>>
>>>>>>>
>>>>>>> But if they returned differnt values, they will have different
>>>>>>> descriptions.
>>>>>>>
>>>>>>> Otherwise, how could a UTM get the right answer, since it only gets
>>>>>>> the description.
>>>>>>
>>>>>> We can get around all of this stuff by simply using this criteria:
>>>>>> Date 10/13/2022 11:29:23 AM
>>>>>> *MIT Professor Michael Sipser agreed this verbatim paragraph is
>>>>>> correct*
>>>>>> (He has neither reviewed nor agreed to anything else in this paper)
>>>>>> (a) If simulating halt decider H correctly simulates its input D
>>>>>> until H
>>>>>> correctly determines that its simulated D would never stop running
>>>>>> unless aborted then
>>>>>> (b) H can abort its simulation of D and correctly report that D
>>>>>> specifies a non-halting sequence of configurations.
>>>>>>
>>>>>> *When we apply this criteria* (elaborated above)
>>>>>> Will you halt if you never abort your simulation?
>>>>>> *Then the halting problem is conquered*
>>>>>>
>>>>>> When two different machines implementing this criteria
>>>>>> get different results from identical inputs then we
>>>>>> know that Pathological Self-Reference has been detected.
>>>>>>
>>>>>> We don't even need to know that for:
>>>>>> *denial-of-service-attack detection*
>>>>>> *NO always means reject as unsafe*
>>>>>>
>>>>>
>>>>> But, Halting theorem never said "there's an input that halts
>>>>> all machines", it just says "for any machine, there's an input
>>>>> that halts it".
>>>>>
>>>>> Where "halt the machine" means "put it in an infinite loop".
>>>>>
>>>>> So, rather, Halting theorem never said "there's an input that
>>>>> exhausts all machines", it just says, "for any machine, there's
>>>>> an input that exhausts it".
>>>>>
>>>>> I still don't see how that would be with infinite tapes though,
>>>>> without a means of checking all the way right the tape in one
>>>>> step, i.e. that it's 1's or 0's or any pattern at all, any
>>>>> input that unbounded with respect to the machine basically
>>>>> exhausts it or where the machine would run in the unbounded.
>>>>>
>>>>>
>>>>> Of course any finite tape, has a static analysis that is
>>>>> not infinite, that decides whether or not it halts
>>>>> (or, loops, or grows, the state space of the decider).
>>>>>
>>>>> Static analysis has to either enumerate or _infer_ the
>>>>> state-space, where equal values in what's determined
>>>>> the idempotent can detect loops, while inequalities
>>>>> or proven divergence, ..., can detect unbounded growth.
>>>>>
>>>>> Now, proving convergence or divergence is its own kind
>>>>> of thing. For example, there are series that converge
>>>>> very slowly, and series that diverge very slowly. These
>>>>> are subtly intractable to analysis.
>>>>>
>>>>> Then the usual idea of progressions that rapidly grow
>>>>> yet return to what's detectable, are pathological to
>>>>> analysis, only in resources not correctness or completion,
>>>>> vis-a-vis the subtle intractability of the convergence or
>>>>> divergence what either halts, or loops, or grows unboundedly.
>>>>>
>>>>> Separately "not-halts" into "loops or grows unboundedly",
>>>>> has that really for most matters of interest of understanding
>>>>> the state-space of programs, is "halts or enters loops"
>>>>> and not "grows unboundedly".
>>>>>
>>>>> I.e. if the Halting problem is basically subject the
>>>>> subtle intractability of slow convergence, that otherwise
>>>>> it can just infer divergence and decide, practically
>>>>> it's sort of more relevant what the machine would be
>>>>> doing on the input on the tape, then with respect to
>>>>> beyond the Turing theory, of the state of the read-head,
>>>>> what happens when somebody modifies the tape, or events,
>>>>> the write-head.
>>>>>
>>>>> Anyways though for bounded inputs, besides slow divergence,
>>>>> it's to be made clear that _most all_ and _almost all_
>>>>> programs _are_ decided their behavior by static analysis.
>>>>>
>>>>> Though, "most all" and "almost all" might be a bit strong,
>>>>> but pretty much all that don't involve "the subtle intractability
>>>>> of slow divergence".
>>>>>
>>>>>
>>>>>
>>>>> Giving the idea that an existence result
>>>>> is in any way the expected result here
>>>>> seems sort of the root of this dilem-na.
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> (Though that the real numbers in ZFC have a well-ordering
>>>>> and if they had a normal ordering that was a well-ordering,
>>>>> that would be a thing, because ZFC has a well-ordering of
>>>>> [0,1], but can't give one.)
>>>>>
>>>>>
>>>>
>>>>
>>>>
>>>> What I'm saying is that you'd need to solve all
>>>> cases of slow convergence and slow divergence
>>>> to have a correct and complete halt decider,
>>>> for the unbounded, and finite, if not the infinite.
>>>>
>>>> Most though setups of convergence though
>>>> would effectively exhaust according to the
>>>> existence of criteria of convergence as modeling
>>>> the computation, though, so, if you exclude
>>>> slow convergence and slow divergence,
>>>> then a subset of very usual machines is of
>>>> course quite all decidable. (Or decide-able,
>>>> or as I put it once, "not deciddable".)
>>>>
>>>>
>>>>
>>>> Well-ordering the reals in ZFC, that's it own
>>>> sort issue - that a well-ordering implies the
>>>> existence of a bijection from ordinals to a
>>>> set, then as that as tuples, subsets of those
>>>> are well-ordered by their ordinal part, then
>>>> that if ever uncountably many of those are
>>>> in normal order their real part and irrationals,
>>>> then between those are rationals. I.e.,
>>>> ZFC doesn't give an example of well-ordering
>>>> the reals, but there is one.
>>>>
>>>>
>>>> So, Peter, I think you're kind of misreading that
>>>> quote you authoritated. It just says "if static
>>>> analysis detects a loop or unboundedly growing
>>>> then it's not-halts".
>>>>
>>>>
>>>> Of course, for any bounded resource, there
>>>> are arbitrarily larger bounded resources
>>>> required by what's called pathological,
>>>> that an arbitrarily larger resource can though solve.
>>>>
>>>>
>>>>
>>>> So anyways "slow convergence" and "slow divergence",
>>>> have that in mathematics, it's unknown for some series
>>>> whether they converge or diverge, though it's usually
>>>> accepted that the inverse powers of 2 converge and
>>>> that the sum of integers diverges, and that periodic
>>>> functions do neither.
>>>>
>>>> I.e. the criteria of convergence and existence of a limit,
>>>> and the special case "diverges as going to infinity",
>>>> are not yet closed in today's mathematics.
>>>>
>>>> It's sort of a conjecture or independent whether
>>>> they ever could be, closed, and complete, the criteria.
>>>>
>>>>
>>>
>>> I have spent 20 years on The conventional halting problem proofs,
>>> Gödel 1931 Incompleteness and the Liar Paradox. I have only spent
>>> about seven years on Tarski Undefinability.
>>>
>>> The Halting problem is the only one of these that has a formal system
>>> that cannot have hidden gaps its its reasoning. If a machine can do it
>>> then it can be done.
>>>
>>
>> "Algorithmics" is a study of algorithms. The most usual
>> consideration is "asymptotics", "asymptotics of complexity",
>> with the idea that computing does work, that there are
>> closures and completions, and there are approximations and
>> attenuations, as it were, in the full and the final and
>> the initial and the partial.
>>
>> The most usual resources are time, and space, that actions
>> occur in time, and according to a structure, in space.
>>
>> Of course any CS grad pretty much knows that.
>>
>>
>> The objects of mathematics, have it so, that the finite
>> and bounded resources of any collection of digital and
>> binary machines, are finite and bounded, while, the unbounded
>> models of mathematics that represent them, are unbounded,
>> it's bad that anybody ever said a Turing tape was "infinite",
>> except that it's unbounded as an object of mathematics, and
>> the model of any finite, digital, binary machine.
>>
>> One might aver "digital" has a complement in computing,
>> "analog", which represents the continuous instead of
>> the discrete, and one might aver "binary" has a complement,
>> either "real-valued" or "fuzzy" or variously about the
>> "multi-valent", in logic, the models, according to the
>> mechanisms, the models, their mechanics, machines.
>>
>> The Turing machine though is a model of unbounded machine
>> as what any thusly simply mechanical model can implement,
>> up to its bounds.
>>
>>
>> So, the objects of mathematics, then involve analytical
>> results. I can't draw a circle, though a compass may
>> portray a diagram arbitrarily close, and no different.
>> It's the analytical results, that make so for why
>> mathematics is: "sensible, fungible, and tractable",
>> and for "the ubiquitous success of mathematics" in
>> all such matters of physics, which is a continuum mechanics,
>> and as well discrete mechanics.
>>
>> Of course, results exist in mathematical objects,
>> after the unbounded the unbounded and complete,
>> which is why closures and completions are so
>> relevant to all such matters, vis-a-vis approximations
>> and attenuations, which in bounded terms result
>> indistinguishably non-different results,
>> up to precision, say, or tolerances.
>>
>> So, "machine" might be discrete, binary, and digital,
>> or it might be continuous, multivalent, and analog.
>>
>> The Turing machine in a concrete form, is bounded.
>>
>> Similar models, here mostly for the stack models,
>> stack machines, among finite-state-machines then
>> for example concepts like "unbounded nondeterminism",
>> here is pretty much about determinism, about the
>> defined behavior of a machine according to a
>> mathematical model that given defined behavior
>> of the mechanisms results a mechanical model of
>> a mathematical model, such a theory as this.
>>
>> So, the theory of computing, with mechanical or
>> computational models, models of computing, is
>> pretty much exactly like the theory of physics,
>> with the attachment of physical models to the
>> mathematical models, and rather, "interpreted as",
>> the models, the models, with respect to each other.
>>
>> I.e. the extensionality that so follows "they're
>> equal, or same, or not different, they represent
>> each other, they're equi-interpretable, they model
>> each other", the models, of logical and mathematical
>> and computational and mechanical and physical models,
>> help represent that over all the entire thing there
>> is a usual theory for the entire thing, it's a theory
>> where model theory models extensionality, and in identity
>> intensionality, about equality, tautology, identity,
>> and sameness/difference, and nearness/farness, and all
>> these usual aspects of the mathematical models'
>> arithmetic, algebra, and geometry. (And function
>> theory and topology, with respect to categories and
>> types, sets and parts, relations and predicates,
>> then all the model-building among that which is
>> all the propositional or the stipulated or axiomatic.)
>>
>> The idea is that this sort of platonistic universe of
>> mathematical objects has always existed, and it's just
>> discovered, then that axiomatics for example, just
>> sort of results a model theory of it, with regards
>> to models, the modular, modulation, modality, and the mode.
>>
>>
>> So, a machine, with respect to computation, establishing
>> the validity of interpretations of models, is subject to
>> filling out all the above, vis-a-vis what it can do,
>> and it can-do.
>>
>>
>> Then, "slowly converging" and "slowly diverging"
>> are examples that get get into that there are more
>> law(s) of large numbers, than the usual classical
>> law of large numbers.
>>
>> Some people variously do or don't have a mathematical
>> model of larger numbers, or the infinite, at all.
>> It's a totally ancient and dogmatic tradition that
>> no, we finite peoples, persons, or agents of limited
>> and finite means, no we do not have discrete, digital,
>> binary mechanical models of the infinite.
>>
>> Yet, it's also about the first thing that deduction
>> arrives at just from the plain contemplation of the
>> beyond the unbounded, like the theories of Democritus
>> and Atomism or the theories of Zeno and Continuity,
>> that make it so we at least have something to arrive
>> at for models of the unbounded, as "methods of exhaustion",
>> the approximation and attenuative what results
>> the asymptotics of algorithmics.
>>
>> So, we do have mental models of the continuous and infinite.
>> And, where physics is a continuum mechanics,
>> there are physical models of it as well, then
>> with regards to the philosophy of science and
>> the scientific method and the Big Science and
>> from the Primary Science, lots to establish
>> that the continuous and analog has mathematical
>> results, as whethe, "sensible, fungible, and tractable",
>> to machines at all.
>>
>> Numerical methods then, and, approximation and
>> attenuation, result from analysis, why they exist,
>> here for the criteria of convergence and existence
>> of limits, in the closed, in the complete.
>>
>>
>> So, a metonymy, unless it's "the Strong Metonymy",
>> which is utter intensionality of "the ground model",
>> is yet a metaphor and a metaphor yet a weaker metonymy
>> joint among weaker metonymys, that, ontology, has
>> that there are or may be a completion, of ontology,
>> as a "strong ontology" and "strong metonymy",
>> but otherwise it's just a bag of facts.
>>
>>
>>
>> Here then there's certainly a perceived analytical
>> development Cantor space, and, Square Cantor space,
>> and, Signal Cantor space as it were, about that among
>> the law(s) of large numbers, there are definitions of
>> continuity, at least including field-reals, line-reals,
>> and signal-reals, three sorts continuous domains.
>>
>>
>> Mathematics owes physics this to better begin to
>> explain, to disclose, to find truths of, continuum mechanics.
>>
>>
>> Then that analog, fuzzy, multi-value, "quantum" computing
>> models (parts) of that, that discrete, binary, digital
>> computing does not, is a thing.
>>
>>
>>
>> The convergence of terms in series is an example
>> of a model of things, that, sometimes has clear
>> and direct and perfectly accurate representations,
>> models, in stack machine, and sometimes doesn't.
>>
>>
>> Of course, for any bounded input, there is a sufficiently
>> larger bounded analyzer, that does solve it. So, why not
>> just put those all together? It would be infinite,
>> yet most things are not. (Or, you know, are.)
>>
>>
>> It's called "foundations" the study of all these things
>> as a fundamental coherency, a thing together, while of
>> its parts.
>>
>> So, students should well have the concepts of "abacus"
>> and "slide rule" and other "tabulators and computers"
>> of the usual manual mechanical variety down, according
>> to the principles of the mathematical models behind them,
>> then introducing the law(s) of large numbers, then
>> introducing the idea of a standard model of an infinite,
>> about rates, related rates, asymptotics, and bounds.
>>
>>
>> "Foundations" then the idea is that it's been around
>> forever, it's our dogma and from our canon and it develops
>> over time, and the current edition is called "modern".
>
> Did I make any mistakes?
>
> Tarski anchors his proof in the Liar Paradox
> https://liarparadox.org/Tarski_247_248.pdf
>
> How Tarski encodes the Liar Paradox
> x ∉ True if and only if p
> where the symbol 'p' represents the whole sentence x
>
> This is transformed into line 01 of his proof
> by replacing True with Provable
>
> *Here is the Tarski Undefinability Theorem proof*
> (1) x ∉ Provable if and only if p // assumption
> (2) x ∈ True if and only if p // assumption
> (3) x ∉ Provable if and only if x ∈ True.
> (4) either x ∉ True or x̄ ∉ True; // axiom: ~True(x) ∨ ~True(~x)
> (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
> (6) if x̄ ∈ Provable, then x̄ ∈ True; // axiom: Provable(~x) → True(~x)
> (7) x ∈ True
> (8) x ∉ Provable
> (9) x̄ ∉ Provable
>
> Tarski's Full proof
> https://liarparadox.org/Tarski_275_276.pdf
>
>
>

On 3/14/2024 11:58 AM, Ross Finlayson wrote:
> On 03/13/2024 10:20 PM, olcott wrote:
>> On 3/13/2024 1:16 PM, Ross Finlayson wrote:
>>> On 03/12/2024 09:00 PM, olcott wrote:
>>>> On 3/12/2024 10:49 PM, Ross Finlayson wrote:
>>>>> On 03/12/2024 08:23 PM, Ross Finlayson wrote:
>>>>>> On 03/12/2024 07:52 PM, olcott wrote:
>>>>>>> On 3/12/2024 9:28 PM, Richard Damon wrote:
>>>>>>>> On 3/12/24 4:31 PM, olcott wrote:
>>>>>>>>> On 3/12/2024 6:11 PM, Richard Damon wrote:
>>>>>>>>>> On 3/12/24 3:53 PM, olcott wrote:
>>>>>>>>>>> On 3/12/2024 5:30 PM, Richard Damon wrote:
>>>>>>>>>>>> On 3/12/24 2:34 PM, olcott wrote:
>>>>>>>>>>>>> On 3/12/2024 4:23 PM, Richard Damon wrote:
>>>>>>>>>>>>>> On 3/12/24 1:11 PM, olcott wrote:
>>>>>>>>>>>>>>> On 3/12/2024 2:40 PM, Richard Damon wrote:
>>>>>>>>>>>>>>>> On 3/12/24 12:02 PM, olcott wrote:
>>>>>>>>>>>>>>>>> On 3/12/2024 1:31 PM, immibis wrote:
>>>>>>>>>>>>>>>>>> On 12/03/24 19:12, olcott wrote:
>>>>>>>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> There is some input TMD to every H such that
>>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> And it can be a different TMD to each H.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> When we disallow decider/input pairs that are incorrect
>>>>>>>>>>>>>>>>>>> questions where both YES and NO are the wrong answer
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> Once we understand that either YES or NO is the right
>>>>>>>>>>>>>>>>>> answer, the whole rebuttal is tossed out as invalid and
>>>>>>>>>>>>>>>>>> incorrect.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩
>>>>>>>>>>>>>>>>> halts
>>>>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn   // Ĥ applied to ⟨Ĥ⟩
>>>>>>>>>>>>>>>>> does
>>>>>>>>>>>>>>>>> not halt
>>>>>>>>>>>>>>>>> BOTH YES AND NO ARE THE WRONG ANSWER FOR EVERY Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> No, because a given H will only go to one of the answers.
>>>>>>>>>>>>>>>> THAT
>>>>>>>>>>>>>>>> will be wrong, and the other one right.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Not exactly. A pair of otherwise identical machines that
>>>>>>>>>>>>>>> (that are contained within the above specified set)
>>>>>>>>>>>>>>> only differ by return value will both be wrong on the
>>>>>>>>>>>>>>> same pathological input.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> You mean a pair of DIFFERENT machines. Any difference is
>>>>>>>>>>>>>> different.
>>>>>>>>>>>>>
>>>>>>>>>>>>> Every decider/input pair (referenced in the above set) has a
>>>>>>>>>>>>> corresponding decider/input pair that only differs by the
>>>>>>>>>>>>> return
>>>>>>>>>>>>> value of its decider.
>>>>>>>>>>>>
>>>>>>>>>>>> Nope.
>>>>>>>>>>>>
>>>>>>>>>>> ∀ H ∈ Turing_Machines_Returning_Boolean
>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>
>>>>>>>>>>> Every H/TMD pair (referenced in the above set) has a
>>>>>>>>>>> corresponding H/TMD pair that only differs by the return
>>>>>>>>>>> value of its Boolean_TM.
>>>>>>>>>>
>>>>>>>>>> That isn't in the set above.
>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> That both of these H/TMD pairs get the wrong answer proves that
>>>>>>>>>>> their question was incorrect because the opposite answer to the
>>>>>>>>>>> same question is also proven to be incorrect.
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>> Nope, since both aren't in the set selected.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> When they are deciders that must get the correct answer both
>>>>>>>>> of them are not in the set.
>>>>>>>>
>>>>>>>> *IF* they are correct decider.
>>>>>>>>
>>>>>>>> WHen we select from all Turing Machine Deciders, there is no
>>>>>>>> requirement that any of them get any particular answer right.
>>>>>>>>
>>>>>>>> So, ALL deciders are in the set that we cycle through and apply the
>>>>>>>> following logic to ALL of them.
>>>>>>>>
>>>>>>>> Each is them paired with an input that it will get wrong, and the
>>>>>>>> existance of the input was what as just proven, the ^ template
>>>>>>>>
>>>>>>>>>
>>>>>>>>> When they are Turing_Machines_Returning_Boolean the this
>>>>>>>>> set inherently includes identical pairs that only differ
>>>>>>>>> by return value.
>>>>>>>>
>>>>>>>> But in the step of select and input that they will get wrong, they
>>>>>>>> will be givne DIFFERENT inputs.
>>>>>>>>
>>>>>>>>>
>>>>>>>>>> You just don't understand what that statement is saying.
>>>>>>>>>>
>>>>>>>>>> I've expalined it, but it seems over you head.
>>>>>>>>>>
>>>>>>>>> No the problem is that you are not paying attention.
>>>>>>>>
>>>>>>>> No, you keep on making STUPID mistakes, like thinking that select a
>>>>>>>> input that the machine will get wrong needs to be the same for two
>>>>>>>> differnt machines.
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>>>
>>>>>>>>>> For Every H, we show we can find at least one input (chosen
>>>>>>>>>> just for
>>>>>>>>>> that machine) that it will get wrong.
>>>>>>>>>>
>>>>>>>>> When we use machine templates then we can see instances of
>>>>>>>>> the same machine that only differs by return value where both
>>>>>>>>> get the wrong answer on the same input. By same input I mean
>>>>>>>>> the same finite string of numerical values.
>>>>>>>>>
>>>>>>>>
>>>>>>>> But if they returned differnt values, they will have different
>>>>>>>> descriptions.
>>>>>>>>
>>>>>>>> Otherwise, how could a UTM get the right answer, since it only gets
>>>>>>>> the description.
>>>>>>>
>>>>>>> We can get around all of this stuff by simply using this criteria:
>>>>>>> Date 10/13/2022 11:29:23 AM
>>>>>>> *MIT Professor Michael Sipser agreed this verbatim paragraph is
>>>>>>> correct*
>>>>>>> (He has neither reviewed nor agreed to anything else in this paper)
>>>>>>> (a) If simulating halt decider H correctly simulates its input D
>>>>>>> until H
>>>>>>> correctly determines that its simulated D would never stop running
>>>>>>> unless aborted then
>>>>>>> (b) H can abort its simulation of D and correctly report that D
>>>>>>> specifies a non-halting sequence of configurations.
>>>>>>>
>>>>>>> *When we apply this criteria* (elaborated above)
>>>>>>> Will you halt if you never abort your simulation?
>>>>>>> *Then the halting problem is conquered*
>>>>>>>
>>>>>>> When two different machines implementing this criteria
>>>>>>> get different results from identical inputs then we
>>>>>>> know that Pathological Self-Reference has been detected.
>>>>>>>
>>>>>>> We don't even need to know that for:
>>>>>>> *denial-of-service-attack detection*
>>>>>>> *NO always means reject as unsafe*
>>>>>>>
>>>>>>
>>>>>> But, Halting theorem never said "there's an input that halts
>>>>>> all machines", it just says "for any machine, there's an input
>>>>>> that halts it".
>>>>>>
>>>>>> Where "halt the machine" means "put it in an infinite loop".
>>>>>>
>>>>>> So, rather, Halting theorem never said "there's an input that
>>>>>> exhausts all machines", it just says, "for any machine, there's
>>>>>> an input that exhausts it".
>>>>>>
>>>>>> I still don't see how that would be with infinite tapes though,
>>>>>> without a means of checking all the way right the tape in one
>>>>>> step, i.e. that it's 1's or 0's or any pattern at all, any
>>>>>> input that unbounded with respect to the machine basically
>>>>>> exhausts it or where the machine would run in the unbounded.
>>>>>>
>>>>>>
>>>>>> Of course any finite tape, has a static analysis that is
>>>>>> not infinite, that decides whether or not it halts
>>>>>> (or, loops, or grows, the state space of the decider).
>>>>>>
>>>>>> Static analysis has to either enumerate or _infer_ the
>>>>>> state-space, where equal values in what's determined
>>>>>> the idempotent can detect loops, while inequalities
>>>>>> or proven divergence, ..., can detect unbounded growth.
>>>>>>
>>>>>> Now, proving convergence or divergence is its own kind
>>>>>> of thing. For example, there are series that converge
>>>>>> very slowly, and series that diverge very slowly. These
>>>>>> are subtly intractable to analysis.
>>>>>>
>>>>>> Then the usual idea of progressions that rapidly grow
>>>>>> yet return to what's detectable, are pathological to
>>>>>> analysis, only in resources not correctness or completion,
>>>>>> vis-a-vis the subtle intractability of the convergence or
>>>>>> divergence what either halts, or loops, or grows unboundedly.
>>>>>>
>>>>>> Separately "not-halts" into "loops or grows unboundedly",
>>>>>> has that really for most matters of interest of understanding
>>>>>> the state-space of programs, is "halts or enters loops"
>>>>>> and not "grows unboundedly".
>>>>>>
>>>>>> I.e. if the Halting problem is basically subject the
>>>>>> subtle intractability of slow convergence, that otherwise
>>>>>> it can just infer divergence and decide, practically
>>>>>> it's sort of more relevant what the machine would be
>>>>>> doing on the input on the tape, then with respect to
>>>>>> beyond the Turing theory, of the state of the read-head,
>>>>>> what happens when somebody modifies the tape, or events,
>>>>>> the write-head.
>>>>>>
>>>>>> Anyways though for bounded inputs, besides slow divergence,
>>>>>> it's to be made clear that _most all_ and _almost all_
>>>>>> programs _are_ decided their behavior by static analysis.
>>>>>>
>>>>>> Though, "most all" and "almost all" might be a bit strong,
>>>>>> but pretty much all that don't involve "the subtle intractability
>>>>>> of slow divergence".
>>>>>>
>>>>>>
>>>>>>
>>>>>> Giving the idea that an existence result
>>>>>> is in any way the expected result here
>>>>>> seems sort of the root of this dilem-na.
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> (Though that the real numbers in ZFC have a well-ordering
>>>>>> and if they had a normal ordering that was a well-ordering,
>>>>>> that would be a thing, because ZFC has a well-ordering of
>>>>>> [0,1], but can't give one.)
>>>>>>
>>>>>>
>>>>>
>>>>>
>>>>>
>>>>> What I'm saying is that you'd need to solve all
>>>>> cases of slow convergence and slow divergence
>>>>> to have a correct and complete halt decider,
>>>>> for the unbounded, and finite, if not the infinite.
>>>>>
>>>>> Most though setups of convergence though
>>>>> would effectively exhaust according to the
>>>>> existence of criteria of convergence as modeling
>>>>> the computation, though, so, if you exclude
>>>>> slow convergence and slow divergence,
>>>>> then a subset of very usual machines is of
>>>>> course quite all decidable.  (Or decide-able,
>>>>> or as I put it once, "not deciddable".)
>>>>>
>>>>>
>>>>>
>>>>> Well-ordering the reals in ZFC, that's it own
>>>>> sort issue - that a well-ordering implies the
>>>>> existence of a bijection from ordinals to a
>>>>> set, then as that as tuples, subsets of those
>>>>> are well-ordered by their ordinal part, then
>>>>> that if ever uncountably many of those are
>>>>> in normal order their real part and irrationals,
>>>>> then between those are rationals.  I.e.,
>>>>> ZFC doesn't give an example of well-ordering
>>>>> the reals, but there is one.
>>>>>
>>>>>
>>>>> So, Peter, I think you're kind of misreading that
>>>>> quote you authoritated.  It just says "if static
>>>>> analysis detects a loop or unboundedly growing
>>>>> then it's not-halts".
>>>>>
>>>>>
>>>>> Of course, for any bounded resource, there
>>>>> are arbitrarily larger bounded resources
>>>>> required by what's called pathological,
>>>>> that an arbitrarily larger resource can though solve.
>>>>>
>>>>>
>>>>>
>>>>> So anyways "slow convergence" and "slow divergence",
>>>>> have that in mathematics, it's unknown for some series
>>>>> whether they converge or diverge, though it's usually
>>>>> accepted that the inverse powers of 2 converge and
>>>>> that the sum of integers diverges, and that periodic
>>>>> functions do neither.
>>>>>
>>>>> I.e. the criteria of convergence and existence of a limit,
>>>>> and the special case "diverges as going to infinity",
>>>>> are not yet closed in today's mathematics.
>>>>>
>>>>> It's sort of a conjecture or independent whether
>>>>> they ever could be, closed, and complete, the criteria.
>>>>>
>>>>>
>>>>
>>>> I have spent 20 years on The conventional halting problem proofs,
>>>> Gödel 1931 Incompleteness and the Liar Paradox. I have only spent
>>>> about seven years on Tarski Undefinability.
>>>>
>>>> The Halting problem is the only one of these that has a formal system
>>>> that cannot have hidden gaps its its reasoning. If a machine can do it
>>>> then it can be done.
>>>>
>>>
>>> "Algorithmics" is a study of algorithms. The most usual
>>> consideration is "asymptotics", "asymptotics of complexity",
>>> with the idea that computing does work, that there are
>>> closures and completions, and there are approximations and
>>> attenuations, as it were, in the full and the final and
>>> the initial and the partial.
>>>
>>> The most usual resources are time, and space, that actions
>>> occur in time, and according to a structure, in space.
>>>
>>> Of course any CS grad pretty much knows that.
>>>
>>>
>>> The objects of mathematics, have it so, that the finite
>>> and bounded resources of any collection of digital and
>>> binary machines, are finite and bounded, while, the unbounded
>>> models of mathematics that represent them, are unbounded,
>>> it's bad that anybody ever said a Turing tape was "infinite",
>>> except that it's unbounded as an object of mathematics, and
>>> the model of any finite, digital, binary machine.
>>>
>>> One might aver "digital" has a complement in computing,
>>> "analog", which represents the continuous instead of
>>> the discrete, and one might aver "binary" has a complement,
>>> either "real-valued" or "fuzzy" or variously about the
>>> "multi-valent", in logic, the models, according to the
>>> mechanisms, the models, their mechanics, machines.
>>>
>>> The Turing machine though is a model of unbounded machine
>>> as what any thusly simply mechanical model can implement,
>>> up to its bounds.
>>>
>>>
>>> So, the objects of mathematics, then involve analytical
>>> results. I can't draw a circle, though a compass may
>>> portray a diagram arbitrarily close, and no different.
>>> It's the analytical results, that make so for why
>>> mathematics is: "sensible, fungible, and tractable",
>>> and for "the ubiquitous success of mathematics" in
>>> all such matters of physics, which is a continuum mechanics,
>>> and as well discrete mechanics.
>>>
>>> Of course, results exist in mathematical objects,
>>> after the unbounded the unbounded and complete,
>>> which is why closures and completions are so
>>> relevant to all such matters, vis-a-vis approximations
>>> and attenuations, which in bounded terms result
>>> indistinguishably non-different results,
>>> up to precision, say, or tolerances.
>>>
>>> So, "machine" might be discrete, binary, and digital,
>>> or it might be continuous, multivalent, and analog.
>>>
>>> The Turing machine in a concrete form, is bounded.
>>>
>>> Similar models, here mostly for the stack models,
>>> stack machines, among finite-state-machines then
>>> for example concepts like "unbounded nondeterminism",
>>> here is pretty much about determinism, about the
>>> defined behavior of a machine according to a
>>> mathematical model that given defined behavior
>>> of the mechanisms results a mechanical model of
>>> a mathematical model, such a theory as this.
>>>
>>> So, the theory of computing, with mechanical or
>>> computational models, models of computing, is
>>> pretty much exactly like the theory of physics,
>>> with the attachment of physical models to the
>>> mathematical models, and rather, "interpreted as",
>>> the models, the models, with respect to each other.
>>>
>>> I.e. the extensionality that so follows "they're
>>> equal, or same, or not different, they represent
>>> each other, they're equi-interpretable, they model
>>> each other", the models, of logical and mathematical
>>> and computational and mechanical and physical models,
>>> help represent that over all the entire thing there
>>> is a usual theory for the entire thing, it's a theory
>>> where model theory models extensionality, and in identity
>>> intensionality, about equality, tautology, identity,
>>> and sameness/difference, and nearness/farness, and all
>>> these usual aspects of the mathematical models'
>>> arithmetic, algebra, and geometry. (And function
>>> theory and topology, with respect to categories and
>>> types, sets and parts, relations and predicates,
>>> then all the model-building among that which is
>>> all the propositional or the stipulated or axiomatic.)
>>>
>>> The idea is that this sort of platonistic universe of
>>> mathematical objects has always existed, and it's just
>>> discovered, then that axiomatics for example, just
>>> sort of results a model theory of it, with regards
>>> to models, the modular, modulation, modality, and the mode.
>>>
>>>
>>> So, a machine, with respect to computation, establishing
>>> the validity of interpretations of models, is subject to
>>> filling out all the above, vis-a-vis what it can do,
>>> and it can-do.
>>>
>>>
>>> Then, "slowly converging" and "slowly diverging"
>>> are examples that get get into that there are more
>>> law(s) of large numbers, than the usual classical
>>> law of large numbers.
>>>
>>> Some people variously do or don't have a mathematical
>>> model of larger numbers, or the infinite, at all.
>>> It's a totally ancient and dogmatic tradition that
>>> no, we finite peoples, persons, or agents of limited
>>> and finite means, no we do not have discrete, digital,
>>> binary mechanical models of the infinite.
>>>
>>> Yet, it's also about the first thing that deduction
>>> arrives at just from the plain contemplation of the
>>> beyond the unbounded, like the theories of Democritus
>>> and Atomism or the theories of Zeno and Continuity,
>>> that make it so we at least have something to arrive
>>> at for models of the unbounded, as "methods of exhaustion",
>>> the approximation and attenuative what results
>>> the asymptotics of algorithmics.
>>>
>>> So, we do have mental models of the continuous and infinite.
>>> And, where physics is a continuum mechanics,
>>> there are physical models of it as well, then
>>> with regards to the philosophy of science and
>>> the scientific method and the Big Science and
>>> from the Primary Science, lots to establish
>>> that the continuous and analog has mathematical
>>> results, as whethe, "sensible, fungible, and tractable",
>>> to machines at all.
>>>
>>> Numerical methods then, and, approximation and
>>> attenuation, result from analysis, why they exist,
>>> here for the criteria of convergence and existence
>>> of limits, in the closed, in the complete.
>>>
>>>
>>> So, a metonymy, unless it's "the Strong Metonymy",
>>> which is utter intensionality of "the ground model",
>>> is yet a metaphor and a metaphor yet a weaker metonymy
>>> joint among weaker metonymys, that, ontology, has
>>> that there are or may be a completion, of ontology,
>>> as a "strong ontology" and "strong metonymy",
>>> but otherwise it's just a bag of facts.
>>>
>>>
>>>
>>> Here then there's certainly a perceived analytical
>>> development Cantor space, and, Square Cantor space,
>>> and, Signal Cantor space as it were, about that among
>>> the law(s) of large numbers, there are definitions of
>>> continuity, at least including field-reals, line-reals,
>>> and signal-reals, three sorts continuous domains.
>>>
>>>
>>> Mathematics owes physics this to better begin to
>>> explain, to disclose, to find truths of, continuum mechanics.
>>>
>>>
>>> Then that analog, fuzzy, multi-value, "quantum" computing
>>> models (parts) of that, that discrete, binary, digital
>>> computing does not, is a thing.
>>>
>>>
>>>
>>> The convergence of terms in series is an example
>>> of a model of things, that, sometimes has clear
>>> and direct and perfectly accurate representations,
>>> models, in stack machine, and sometimes doesn't.
>>>
>>>
>>> Of course, for any bounded input, there is a sufficiently
>>> larger bounded analyzer, that does solve it. So, why not
>>> just put those all together? It would be infinite,
>>> yet most things are not. (Or, you know, are.)
>>>
>>>
>>> It's called "foundations" the study of all these things
>>> as a fundamental coherency, a thing together, while of
>>> its parts.
>>>
>>> So, students should well have the concepts of "abacus"
>>> and "slide rule" and other "tabulators and computers"
>>> of the usual manual mechanical variety down, according
>>> to the principles of the mathematical models behind them,
>>> then introducing the law(s) of large numbers, then
>>> introducing the idea of a standard model of an infinite,
>>> about rates, related rates, asymptotics, and bounds.
>>>
>>>
>>> "Foundations" then the idea is that it's been around
>>> forever, it's our dogma and from our canon and it develops
>>> over time, and the current edition is called "modern".
>>
>> Did I make any mistakes?
>>
>> Tarski anchors his proof in the Liar Paradox
>> https://liarparadox.org/Tarski_247_248.pdf
>>
>> How Tarski encodes the Liar Paradox
>> x ∉ True if and only if p
>> where the symbol 'p' represents the whole sentence x
>>
>> This is transformed into line 01 of his proof
>> by replacing True with Provable
>>
>> *Here is the Tarski Undefinability Theorem proof*
>> (1) x ∉ Provable if and only if p   // assumption
>> (2) x ∈ True if and only if p       // assumption
>> (3) x ∉ Provable if and only if x ∈ True.
>> (4) either x ∉ True or x̄ ∉ True;    // axiom: ~True(x) ∨ ~True(~x)
>> (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
>> (6) if x̄ ∈ Provable, then x̄ ∈ True;  // axiom: Provable(~x) → True(~x)
>> (7) x ∈ True
>> (8) x ∉ Provable
>> (9) x̄ ∉ Provable
>>
>> Tarski's Full proof
>> https://liarparadox.org/Tarski_275_276.pdf
>>
>>
>>
>
>
> Well I have some collected works of Tarski so
> I'll be looking to them, the excerpt there you
> reference basically talks about the model under
> consideration, as a fragment, of "the meta-theory",
> i.e., "Tarski's meta-theory" is "rather underdefined",
> that what he does is that he says that in the meta-theory,
> that there's something about the model under consideration,
> that isn't true in the model but is true, and he invokes
> Goedel to basically says that Goedel's incompleteness
> applies to this model, which of course must thusly by
> "strong enough to support arithmetic", where otherwise
> of course, Goedel's incompleteness does _not_ apply
> to theories that may be entirely closed categories.
>
> It seems the issue is that you think that you have
> entirely closed categories, in your meta-theory,
> then are applying this meta-theory to another model
> under consideration, which isn't, so it seems like
> you have implicits on your model, which are underdefined,
> rather as the "properly logical" or "non-logical" are
> with respect to being modeled by the logical, that
> you have the some sort of "un-logical" that are the
> result of that the axioms are un-logical, with regards
> to that your "meta-theory" is logical and your model
> under consideration, its _logical_ objects, are actually
> "un-logical" objects with respect to your, "meta-theory".
>
>
> That's a problem overall with "meta-theory", here the
> goal is to arrive at a theory that is its own meta-theory,
> that's also all one language, with regards to otherwise

On 3/14/24 12:01 PM, olcott wrote:
> On 3/14/2024 11:58 AM, Ross Finlayson wrote:
>> On 03/13/2024 10:20 PM, olcott wrote:
>>> On 3/13/2024 1:16 PM, Ross Finlayson wrote:
>>>> On 03/12/2024 09:00 PM, olcott wrote:
>>>>> On 3/12/2024 10:49 PM, Ross Finlayson wrote:
>>>>>> On 03/12/2024 08:23 PM, Ross Finlayson wrote:
>>>>>>> On 03/12/2024 07:52 PM, olcott wrote:
>>>>>>>> On 3/12/2024 9:28 PM, Richard Damon wrote:
>>>>>>>>> On 3/12/24 4:31 PM, olcott wrote:
>>>>>>>>>> On 3/12/2024 6:11 PM, Richard Damon wrote:
>>>>>>>>>>> On 3/12/24 3:53 PM, olcott wrote:
>>>>>>>>>>>> On 3/12/2024 5:30 PM, Richard Damon wrote:
>>>>>>>>>>>>> On 3/12/24 2:34 PM, olcott wrote:
>>>>>>>>>>>>>> On 3/12/2024 4:23 PM, Richard Damon wrote:
>>>>>>>>>>>>>>> On 3/12/24 1:11 PM, olcott wrote:
>>>>>>>>>>>>>>>> On 3/12/2024 2:40 PM, Richard Damon wrote:
>>>>>>>>>>>>>>>>> On 3/12/24 12:02 PM, olcott wrote:
>>>>>>>>>>>>>>>>>> On 3/12/2024 1:31 PM, immibis wrote:
>>>>>>>>>>>>>>>>>>> On 12/03/24 19:12, olcott wrote:
>>>>>>>>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> There is some input TMD to every H such that
>>>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> And it can be a different TMD to each H.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> When we disallow decider/input pairs that are incorrect
>>>>>>>>>>>>>>>>>>>> questions where both YES and NO are the wrong answer
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Once we understand that either YES or NO is the right
>>>>>>>>>>>>>>>>>>> answer, the whole rebuttal is tossed out as invalid and
>>>>>>>>>>>>>>>>>>> incorrect.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩
>>>>>>>>>>>>>>>>>> halts
>>>>>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn   // Ĥ applied to ⟨Ĥ⟩
>>>>>>>>>>>>>>>>>> does
>>>>>>>>>>>>>>>>>> not halt
>>>>>>>>>>>>>>>>>> BOTH YES AND NO ARE THE WRONG ANSWER FOR EVERY Ĥ.H ⟨Ĥ⟩
>>>>>>>>>>>>>>>>>> ⟨Ĥ⟩
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> No, because a given H will only go to one of the answers.
>>>>>>>>>>>>>>>>> THAT
>>>>>>>>>>>>>>>>> will be wrong, and the other one right.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Not exactly. A pair of otherwise identical machines that
>>>>>>>>>>>>>>>> (that are contained within the above specified set)
>>>>>>>>>>>>>>>> only differ by return value will both be wrong on the
>>>>>>>>>>>>>>>> same pathological input.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> You mean a pair of DIFFERENT machines. Any difference is
>>>>>>>>>>>>>>> different.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Every decider/input pair (referenced in the above set) has a
>>>>>>>>>>>>>> corresponding decider/input pair that only differs by the
>>>>>>>>>>>>>> return
>>>>>>>>>>>>>> value of its decider.
>>>>>>>>>>>>>
>>>>>>>>>>>>> Nope.
>>>>>>>>>>>>>
>>>>>>>>>>>> ∀ H ∈ Turing_Machines_Returning_Boolean
>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>
>>>>>>>>>>>> Every H/TMD pair (referenced in the above set) has a
>>>>>>>>>>>> corresponding H/TMD pair that only differs by the return
>>>>>>>>>>>> value of its Boolean_TM.
>>>>>>>>>>>
>>>>>>>>>>> That isn't in the set above.
>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> That both of these H/TMD pairs get the wrong answer proves that
>>>>>>>>>>>> their question was incorrect because the opposite answer to the
>>>>>>>>>>>> same question is also proven to be incorrect.
>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>> Nope, since both aren't in the set selected.
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> When they are deciders that must get the correct answer both
>>>>>>>>>> of them are not in the set.
>>>>>>>>>
>>>>>>>>> *IF* they are correct decider.
>>>>>>>>>
>>>>>>>>> WHen we select from all Turing Machine Deciders, there is no
>>>>>>>>> requirement that any of them get any particular answer right.
>>>>>>>>>
>>>>>>>>> So, ALL deciders are in the set that we cycle through and apply
>>>>>>>>> the
>>>>>>>>> following logic to ALL of them.
>>>>>>>>>
>>>>>>>>> Each is them paired with an input that it will get wrong, and the
>>>>>>>>> existance of the input was what as just proven, the ^ template
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> When they are Turing_Machines_Returning_Boolean the this
>>>>>>>>>> set inherently includes identical pairs that only differ
>>>>>>>>>> by return value.
>>>>>>>>>
>>>>>>>>> But in the step of select and input that they will get wrong, they
>>>>>>>>> will be givne DIFFERENT inputs.
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>> You just don't understand what that statement is saying.
>>>>>>>>>>>
>>>>>>>>>>> I've expalined it, but it seems over you head.
>>>>>>>>>>>
>>>>>>>>>> No the problem is that you are not paying attention.
>>>>>>>>>
>>>>>>>>> No, you keep on making STUPID mistakes, like thinking that
>>>>>>>>> select a
>>>>>>>>> input that the machine will get wrong needs to be the same for two
>>>>>>>>> differnt machines.
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>> For Every H, we show we can find at least one input (chosen
>>>>>>>>>>> just for
>>>>>>>>>>> that machine) that it will get wrong.
>>>>>>>>>>>
>>>>>>>>>> When we use machine templates then we can see instances of
>>>>>>>>>> the same machine that only differs by return value where both
>>>>>>>>>> get the wrong answer on the same input. By same input I mean
>>>>>>>>>> the same finite string of numerical values.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> But if they returned differnt values, they will have different
>>>>>>>>> descriptions.
>>>>>>>>>
>>>>>>>>> Otherwise, how could a UTM get the right answer, since it only
>>>>>>>>> gets
>>>>>>>>> the description.
>>>>>>>>
>>>>>>>> We can get around all of this stuff by simply using this criteria:
>>>>>>>> Date 10/13/2022 11:29:23 AM
>>>>>>>> *MIT Professor Michael Sipser agreed this verbatim paragraph is
>>>>>>>> correct*
>>>>>>>> (He has neither reviewed nor agreed to anything else in this paper)
>>>>>>>> (a) If simulating halt decider H correctly simulates its input D
>>>>>>>> until H
>>>>>>>> correctly determines that its simulated D would never stop running
>>>>>>>> unless aborted then
>>>>>>>> (b) H can abort its simulation of D and correctly report that D
>>>>>>>> specifies a non-halting sequence of configurations.
>>>>>>>>
>>>>>>>> *When we apply this criteria* (elaborated above)
>>>>>>>> Will you halt if you never abort your simulation?
>>>>>>>> *Then the halting problem is conquered*
>>>>>>>>
>>>>>>>> When two different machines implementing this criteria
>>>>>>>> get different results from identical inputs then we
>>>>>>>> know that Pathological Self-Reference has been detected.
>>>>>>>>
>>>>>>>> We don't even need to know that for:
>>>>>>>> *denial-of-service-attack detection*
>>>>>>>> *NO always means reject as unsafe*
>>>>>>>>
>>>>>>>
>>>>>>> But, Halting theorem never said "there's an input that halts
>>>>>>> all machines", it just says "for any machine, there's an input
>>>>>>> that halts it".
>>>>>>>
>>>>>>> Where "halt the machine" means "put it in an infinite loop".
>>>>>>>
>>>>>>> So, rather, Halting theorem never said "there's an input that
>>>>>>> exhausts all machines", it just says, "for any machine, there's
>>>>>>> an input that exhausts it".
>>>>>>>
>>>>>>> I still don't see how that would be with infinite tapes though,
>>>>>>> without a means of checking all the way right the tape in one
>>>>>>> step, i.e. that it's 1's or 0's or any pattern at all, any
>>>>>>> input that unbounded with respect to the machine basically
>>>>>>> exhausts it or where the machine would run in the unbounded.
>>>>>>>
>>>>>>>
>>>>>>> Of course any finite tape, has a static analysis that is
>>>>>>> not infinite, that decides whether or not it halts
>>>>>>> (or, loops, or grows, the state space of the decider).
>>>>>>>
>>>>>>> Static analysis has to either enumerate or _infer_ the
>>>>>>> state-space, where equal values in what's determined
>>>>>>> the idempotent can detect loops, while inequalities
>>>>>>> or proven divergence, ..., can detect unbounded growth.
>>>>>>>
>>>>>>> Now, proving convergence or divergence is its own kind
>>>>>>> of thing. For example, there are series that converge
>>>>>>> very slowly, and series that diverge very slowly. These
>>>>>>> are subtly intractable to analysis.
>>>>>>>
>>>>>>> Then the usual idea of progressions that rapidly grow
>>>>>>> yet return to what's detectable, are pathological to
>>>>>>> analysis, only in resources not correctness or completion,
>>>>>>> vis-a-vis the subtle intractability of the convergence or
>>>>>>> divergence what either halts, or loops, or grows unboundedly.
>>>>>>>
>>>>>>> Separately "not-halts" into "loops or grows unboundedly",
>>>>>>> has that really for most matters of interest of understanding
>>>>>>> the state-space of programs, is "halts or enters loops"
>>>>>>> and not "grows unboundedly".
>>>>>>>
>>>>>>> I.e. if the Halting problem is basically subject the
>>>>>>> subtle intractability of slow convergence, that otherwise
>>>>>>> it can just infer divergence and decide, practically
>>>>>>> it's sort of more relevant what the machine would be
>>>>>>> doing on the input on the tape, then with respect to
>>>>>>> beyond the Turing theory, of the state of the read-head,
>>>>>>> what happens when somebody modifies the tape, or events,
>>>>>>> the write-head.
>>>>>>>
>>>>>>> Anyways though for bounded inputs, besides slow divergence,
>>>>>>> it's to be made clear that _most all_ and _almost all_
>>>>>>> programs _are_ decided their behavior by static analysis.
>>>>>>>
>>>>>>> Though, "most all" and "almost all" might be a bit strong,
>>>>>>> but pretty much all that don't involve "the subtle intractability
>>>>>>> of slow divergence".
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> Giving the idea that an existence result
>>>>>>> is in any way the expected result here
>>>>>>> seems sort of the root of this dilem-na.
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> (Though that the real numbers in ZFC have a well-ordering
>>>>>>> and if they had a normal ordering that was a well-ordering,
>>>>>>> that would be a thing, because ZFC has a well-ordering of
>>>>>>> [0,1], but can't give one.)
>>>>>>>
>>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> What I'm saying is that you'd need to solve all
>>>>>> cases of slow convergence and slow divergence
>>>>>> to have a correct and complete halt decider,
>>>>>> for the unbounded, and finite, if not the infinite.
>>>>>>
>>>>>> Most though setups of convergence though
>>>>>> would effectively exhaust according to the
>>>>>> existence of criteria of convergence as modeling
>>>>>> the computation, though, so, if you exclude
>>>>>> slow convergence and slow divergence,
>>>>>> then a subset of very usual machines is of
>>>>>> course quite all decidable.  (Or decide-able,
>>>>>> or as I put it once, "not deciddable".)
>>>>>>
>>>>>>
>>>>>>
>>>>>> Well-ordering the reals in ZFC, that's it own
>>>>>> sort issue - that a well-ordering implies the
>>>>>> existence of a bijection from ordinals to a
>>>>>> set, then as that as tuples, subsets of those
>>>>>> are well-ordered by their ordinal part, then
>>>>>> that if ever uncountably many of those are
>>>>>> in normal order their real part and irrationals,
>>>>>> then between those are rationals.  I.e.,
>>>>>> ZFC doesn't give an example of well-ordering
>>>>>> the reals, but there is one.
>>>>>>
>>>>>>
>>>>>> So, Peter, I think you're kind of misreading that
>>>>>> quote you authoritated.  It just says "if static
>>>>>> analysis detects a loop or unboundedly growing
>>>>>> then it's not-halts".
>>>>>>
>>>>>>
>>>>>> Of course, for any bounded resource, there
>>>>>> are arbitrarily larger bounded resources
>>>>>> required by what's called pathological,
>>>>>> that an arbitrarily larger resource can though solve.
>>>>>>
>>>>>>
>>>>>>
>>>>>> So anyways "slow convergence" and "slow divergence",
>>>>>> have that in mathematics, it's unknown for some series
>>>>>> whether they converge or diverge, though it's usually
>>>>>> accepted that the inverse powers of 2 converge and
>>>>>> that the sum of integers diverges, and that periodic
>>>>>> functions do neither.
>>>>>>
>>>>>> I.e. the criteria of convergence and existence of a limit,
>>>>>> and the special case "diverges as going to infinity",
>>>>>> are not yet closed in today's mathematics.
>>>>>>
>>>>>> It's sort of a conjecture or independent whether
>>>>>> they ever could be, closed, and complete, the criteria.
>>>>>>
>>>>>>
>>>>>
>>>>> I have spent 20 years on The conventional halting problem proofs,
>>>>> Gödel 1931 Incompleteness and the Liar Paradox. I have only spent
>>>>> about seven years on Tarski Undefinability.
>>>>>
>>>>> The Halting problem is the only one of these that has a formal system
>>>>> that cannot have hidden gaps its its reasoning. If a machine can do it
>>>>> then it can be done.
>>>>>
>>>>
>>>> "Algorithmics" is a study of algorithms. The most usual
>>>> consideration is "asymptotics", "asymptotics of complexity",
>>>> with the idea that computing does work, that there are
>>>> closures and completions, and there are approximations and
>>>> attenuations, as it were, in the full and the final and
>>>> the initial and the partial.
>>>>
>>>> The most usual resources are time, and space, that actions
>>>> occur in time, and according to a structure, in space.
>>>>
>>>> Of course any CS grad pretty much knows that.
>>>>
>>>>
>>>> The objects of mathematics, have it so, that the finite
>>>> and bounded resources of any collection of digital and
>>>> binary machines, are finite and bounded, while, the unbounded
>>>> models of mathematics that represent them, are unbounded,
>>>> it's bad that anybody ever said a Turing tape was "infinite",
>>>> except that it's unbounded as an object of mathematics, and
>>>> the model of any finite, digital, binary machine.
>>>>
>>>> One might aver "digital" has a complement in computing,
>>>> "analog", which represents the continuous instead of
>>>> the discrete, and one might aver "binary" has a complement,
>>>> either "real-valued" or "fuzzy" or variously about the
>>>> "multi-valent", in logic, the models, according to the
>>>> mechanisms, the models, their mechanics, machines.
>>>>
>>>> The Turing machine though is a model of unbounded machine
>>>> as what any thusly simply mechanical model can implement,
>>>> up to its bounds.
>>>>
>>>>
>>>> So, the objects of mathematics, then involve analytical
>>>> results. I can't draw a circle, though a compass may
>>>> portray a diagram arbitrarily close, and no different.
>>>> It's the analytical results, that make so for why
>>>> mathematics is: "sensible, fungible, and tractable",
>>>> and for "the ubiquitous success of mathematics" in
>>>> all such matters of physics, which is a continuum mechanics,
>>>> and as well discrete mechanics.
>>>>
>>>> Of course, results exist in mathematical objects,
>>>> after the unbounded the unbounded and complete,
>>>> which is why closures and completions are so
>>>> relevant to all such matters, vis-a-vis approximations
>>>> and attenuations, which in bounded terms result
>>>> indistinguishably non-different results,
>>>> up to precision, say, or tolerances.
>>>>
>>>> So, "machine" might be discrete, binary, and digital,
>>>> or it might be continuous, multivalent, and analog.
>>>>
>>>> The Turing machine in a concrete form, is bounded.
>>>>
>>>> Similar models, here mostly for the stack models,
>>>> stack machines, among finite-state-machines then
>>>> for example concepts like "unbounded nondeterminism",
>>>> here is pretty much about determinism, about the
>>>> defined behavior of a machine according to a
>>>> mathematical model that given defined behavior
>>>> of the mechanisms results a mechanical model of
>>>> a mathematical model, such a theory as this.
>>>>
>>>> So, the theory of computing, with mechanical or
>>>> computational models, models of computing, is
>>>> pretty much exactly like the theory of physics,
>>>> with the attachment of physical models to the
>>>> mathematical models, and rather, "interpreted as",
>>>> the models, the models, with respect to each other.
>>>>
>>>> I.e. the extensionality that so follows "they're
>>>> equal, or same, or not different, they represent
>>>> each other, they're equi-interpretable, they model
>>>> each other", the models, of logical and mathematical
>>>> and computational and mechanical and physical models,
>>>> help represent that over all the entire thing there
>>>> is a usual theory for the entire thing, it's a theory
>>>> where model theory models extensionality, and in identity
>>>> intensionality, about equality, tautology, identity,
>>>> and sameness/difference, and nearness/farness, and all
>>>> these usual aspects of the mathematical models'
>>>> arithmetic, algebra, and geometry. (And function
>>>> theory and topology, with respect to categories and
>>>> types, sets and parts, relations and predicates,
>>>> then all the model-building among that which is
>>>> all the propositional or the stipulated or axiomatic.)
>>>>
>>>> The idea is that this sort of platonistic universe of
>>>> mathematical objects has always existed, and it's just
>>>> discovered, then that axiomatics for example, just
>>>> sort of results a model theory of it, with regards
>>>> to models, the modular, modulation, modality, and the mode.
>>>>
>>>>
>>>> So, a machine, with respect to computation, establishing
>>>> the validity of interpretations of models, is subject to
>>>> filling out all the above, vis-a-vis what it can do,
>>>> and it can-do.
>>>>
>>>>
>>>> Then, "slowly converging" and "slowly diverging"
>>>> are examples that get get into that there are more
>>>> law(s) of large numbers, than the usual classical
>>>> law of large numbers.
>>>>
>>>> Some people variously do or don't have a mathematical
>>>> model of larger numbers, or the infinite, at all.
>>>> It's a totally ancient and dogmatic tradition that
>>>> no, we finite peoples, persons, or agents of limited
>>>> and finite means, no we do not have discrete, digital,
>>>> binary mechanical models of the infinite.
>>>>
>>>> Yet, it's also about the first thing that deduction
>>>> arrives at just from the plain contemplation of the
>>>> beyond the unbounded, like the theories of Democritus
>>>> and Atomism or the theories of Zeno and Continuity,
>>>> that make it so we at least have something to arrive
>>>> at for models of the unbounded, as "methods of exhaustion",
>>>> the approximation and attenuative what results
>>>> the asymptotics of algorithmics.
>>>>
>>>> So, we do have mental models of the continuous and infinite.
>>>> And, where physics is a continuum mechanics,
>>>> there are physical models of it as well, then
>>>> with regards to the philosophy of science and
>>>> the scientific method and the Big Science and
>>>> from the Primary Science, lots to establish
>>>> that the continuous and analog has mathematical
>>>> results, as whethe, "sensible, fungible, and tractable",
>>>> to machines at all.
>>>>
>>>> Numerical methods then, and, approximation and
>>>> attenuation, result from analysis, why they exist,
>>>> here for the criteria of convergence and existence
>>>> of limits, in the closed, in the complete.
>>>>
>>>>
>>>> So, a metonymy, unless it's "the Strong Metonymy",
>>>> which is utter intensionality of "the ground model",
>>>> is yet a metaphor and a metaphor yet a weaker metonymy
>>>> joint among weaker metonymys, that, ontology, has
>>>> that there are or may be a completion, of ontology,
>>>> as a "strong ontology" and "strong metonymy",
>>>> but otherwise it's just a bag of facts.
>>>>
>>>>
>>>>
>>>> Here then there's certainly a perceived analytical
>>>> development Cantor space, and, Square Cantor space,
>>>> and, Signal Cantor space as it were, about that among
>>>> the law(s) of large numbers, there are definitions of
>>>> continuity, at least including field-reals, line-reals,
>>>> and signal-reals, three sorts continuous domains.
>>>>
>>>>
>>>> Mathematics owes physics this to better begin to
>>>> explain, to disclose, to find truths of, continuum mechanics.
>>>>
>>>>
>>>> Then that analog, fuzzy, multi-value, "quantum" computing
>>>> models (parts) of that, that discrete, binary, digital
>>>> computing does not, is a thing.
>>>>
>>>>
>>>>
>>>> The convergence of terms in series is an example
>>>> of a model of things, that, sometimes has clear
>>>> and direct and perfectly accurate representations,
>>>> models, in stack machine, and sometimes doesn't.
>>>>
>>>>
>>>> Of course, for any bounded input, there is a sufficiently
>>>> larger bounded analyzer, that does solve it. So, why not
>>>> just put those all together? It would be infinite,
>>>> yet most things are not. (Or, you know, are.)
>>>>
>>>>
>>>> It's called "foundations" the study of all these things
>>>> as a fundamental coherency, a thing together, while of
>>>> its parts.
>>>>
>>>> So, students should well have the concepts of "abacus"
>>>> and "slide rule" and other "tabulators and computers"
>>>> of the usual manual mechanical variety down, according
>>>> to the principles of the mathematical models behind them,
>>>> then introducing the law(s) of large numbers, then
>>>> introducing the idea of a standard model of an infinite,
>>>> about rates, related rates, asymptotics, and bounds.
>>>>
>>>>
>>>> "Foundations" then the idea is that it's been around
>>>> forever, it's our dogma and from our canon and it develops
>>>> over time, and the current edition is called "modern".
>>>
>>> Did I make any mistakes?
>>>
>>> Tarski anchors his proof in the Liar Paradox
>>> https://liarparadox.org/Tarski_247_248.pdf
>>>
>>> How Tarski encodes the Liar Paradox
>>> x ∉ True if and only if p
>>> where the symbol 'p' represents the whole sentence x
>>>
>>> This is transformed into line 01 of his proof
>>> by replacing True with Provable
>>>
>>> *Here is the Tarski Undefinability Theorem proof*
>>> (1) x ∉ Provable if and only if p   // assumption
>>> (2) x ∈ True if and only if p       // assumption
>>> (3) x ∉ Provable if and only if x ∈ True.
>>> (4) either x ∉ True or x̄ ∉ True;    // axiom: ~True(x) ∨ ~True(~x)
>>> (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
>>> (6) if x̄ ∈ Provable, then x̄ ∈ True;  // axiom: Provable(~x) → True(~x)
>>> (7) x ∈ True
>>> (8) x ∉ Provable
>>> (9) x̄ ∉ Provable
>>>
>>> Tarski's Full proof
>>> https://liarparadox.org/Tarski_275_276.pdf
>>>
>>>
>>>
>>
>>
>> Well I have some collected works of Tarski so
>> I'll be looking to them, the excerpt there you
>> reference basically talks about the model under
>> consideration, as a fragment, of "the meta-theory",
>> i.e., "Tarski's meta-theory" is "rather underdefined",
>> that what he does is that he says that in the meta-theory,
>> that there's something about the model under consideration,
>> that isn't true in the model but is true, and he invokes
>> Goedel to basically says that Goedel's incompleteness
>> applies to this model, which of course must thusly by
>> "strong enough to support arithmetic", where otherwise
>> of course, Goedel's incompleteness does _not_ apply
>> to theories that may be entirely closed categories.
>>
>> It seems the issue is that you think that you have
>> entirely closed categories, in your meta-theory,
>> then are applying this meta-theory to another model
>> under consideration, which isn't, so it seems like
>> you have implicits on your model, which are underdefined,
>> rather as the "properly logical" or "non-logical" are
>> with respect to being modeled by the logical, that
>> you have the some sort of "un-logical" that are the
>> result of that the axioms are un-logical, with regards
>> to that your "meta-theory" is logical and your model
>> under consideration, its _logical_ objects, are actually
>> "un-logical" objects with respect to your, "meta-theory".
>>
>>
>> That's a problem overall with "meta-theory", here the
>> goal is to arrive at a theory that is its own meta-theory,
>> that's also all one language, with regards to otherwise
>
> Yes that is exactly correct.
>
> *True(L,x) returns TRUE when x is True otherwise returns FALSE*

On 3/14/2024 5:01 PM, Richard Damon wrote:
> On 3/14/24 12:01 PM, olcott wrote:
>> On 3/14/2024 11:58 AM, Ross Finlayson wrote:
>>> On 03/13/2024 10:20 PM, olcott wrote:
>>>> On 3/13/2024 1:16 PM, Ross Finlayson wrote:
>>>>> On 03/12/2024 09:00 PM, olcott wrote:
>>>>>> On 3/12/2024 10:49 PM, Ross Finlayson wrote:
>>>>>>> On 03/12/2024 08:23 PM, Ross Finlayson wrote:
>>>>>>>> On 03/12/2024 07:52 PM, olcott wrote:
>>>>>>>>> On 3/12/2024 9:28 PM, Richard Damon wrote:
>>>>>>>>>> On 3/12/24 4:31 PM, olcott wrote:
>>>>>>>>>>> On 3/12/2024 6:11 PM, Richard Damon wrote:
>>>>>>>>>>>> On 3/12/24 3:53 PM, olcott wrote:
>>>>>>>>>>>>> On 3/12/2024 5:30 PM, Richard Damon wrote:
>>>>>>>>>>>>>> On 3/12/24 2:34 PM, olcott wrote:
>>>>>>>>>>>>>>> On 3/12/2024 4:23 PM, Richard Damon wrote:
>>>>>>>>>>>>>>>> On 3/12/24 1:11 PM, olcott wrote:
>>>>>>>>>>>>>>>>> On 3/12/2024 2:40 PM, Richard Damon wrote:
>>>>>>>>>>>>>>>>>> On 3/12/24 12:02 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>> On 3/12/2024 1:31 PM, immibis wrote:
>>>>>>>>>>>>>>>>>>>> On 12/03/24 19:12, olcott wrote:
>>>>>>>>>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> There is some input TMD to every H such that
>>>>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> And it can be a different TMD to each H.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> When we disallow decider/input pairs that are
>>>>>>>>>>>>>>>>>>>>> incorrect
>>>>>>>>>>>>>>>>>>>>> questions where both YES and NO are the wrong answer
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> Once we understand that either YES or NO is the right
>>>>>>>>>>>>>>>>>>>> answer, the whole rebuttal is tossed out as invalid and
>>>>>>>>>>>>>>>>>>>> incorrect.
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩
>>>>>>>>>>>>>>>>>>> halts
>>>>>>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn   // Ĥ applied to ⟨Ĥ⟩
>>>>>>>>>>>>>>>>>>> does
>>>>>>>>>>>>>>>>>>> not halt
>>>>>>>>>>>>>>>>>>> BOTH YES AND NO ARE THE WRONG ANSWER FOR EVERY Ĥ.H
>>>>>>>>>>>>>>>>>>> ⟨Ĥ⟩ ⟨Ĥ⟩
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> No, because a given H will only go to one of the answers.
>>>>>>>>>>>>>>>>>> THAT
>>>>>>>>>>>>>>>>>> will be wrong, and the other one right.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Not exactly. A pair of otherwise identical machines that
>>>>>>>>>>>>>>>>> (that are contained within the above specified set)
>>>>>>>>>>>>>>>>> only differ by return value will both be wrong on the
>>>>>>>>>>>>>>>>> same pathological input.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> You mean a pair of DIFFERENT machines. Any difference is
>>>>>>>>>>>>>>>> different.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Every decider/input pair (referenced in the above set) has a
>>>>>>>>>>>>>>> corresponding decider/input pair that only differs by the
>>>>>>>>>>>>>>> return
>>>>>>>>>>>>>>> value of its decider.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Nope.
>>>>>>>>>>>>>>
>>>>>>>>>>>>> ∀ H ∈ Turing_Machines_Returning_Boolean
>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>
>>>>>>>>>>>>> Every H/TMD pair (referenced in the above set) has a
>>>>>>>>>>>>> corresponding H/TMD pair that only differs by the return
>>>>>>>>>>>>> value of its Boolean_TM.
>>>>>>>>>>>>
>>>>>>>>>>>> That isn't in the set above.
>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> That both of these H/TMD pairs get the wrong answer proves
>>>>>>>>>>>>> that
>>>>>>>>>>>>> their question was incorrect because the opposite answer to
>>>>>>>>>>>>> the
>>>>>>>>>>>>> same question is also proven to be incorrect.
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>> Nope, since both aren't in the set selected.
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> When they are deciders that must get the correct answer both
>>>>>>>>>>> of them are not in the set.
>>>>>>>>>>
>>>>>>>>>> *IF* they are correct decider.
>>>>>>>>>>
>>>>>>>>>> WHen we select from all Turing Machine Deciders, there is no
>>>>>>>>>> requirement that any of them get any particular answer right.
>>>>>>>>>>
>>>>>>>>>> So, ALL deciders are in the set that we cycle through and
>>>>>>>>>> apply the
>>>>>>>>>> following logic to ALL of them.
>>>>>>>>>>
>>>>>>>>>> Each is them paired with an input that it will get wrong, and the
>>>>>>>>>> existance of the input was what as just proven, the ^ template
>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> When they are Turing_Machines_Returning_Boolean the this
>>>>>>>>>>> set inherently includes identical pairs that only differ
>>>>>>>>>>> by return value.
>>>>>>>>>>
>>>>>>>>>> But in the step of select and input that they will get wrong,
>>>>>>>>>> they
>>>>>>>>>> will be givne DIFFERENT inputs.
>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>> You just don't understand what that statement is saying.
>>>>>>>>>>>>
>>>>>>>>>>>> I've expalined it, but it seems over you head.
>>>>>>>>>>>>
>>>>>>>>>>> No the problem is that you are not paying attention.
>>>>>>>>>>
>>>>>>>>>> No, you keep on making STUPID mistakes, like thinking that
>>>>>>>>>> select a
>>>>>>>>>> input that the machine will get wrong needs to be the same for
>>>>>>>>>> two
>>>>>>>>>> differnt machines.
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>> For Every H, we show we can find at least one input (chosen
>>>>>>>>>>>> just for
>>>>>>>>>>>> that machine) that it will get wrong.
>>>>>>>>>>>>
>>>>>>>>>>> When we use machine templates then we can see instances of
>>>>>>>>>>> the same machine that only differs by return value where both
>>>>>>>>>>> get the wrong answer on the same input. By same input I mean
>>>>>>>>>>> the same finite string of numerical values.
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> But if they returned differnt values, they will have different
>>>>>>>>>> descriptions.
>>>>>>>>>>
>>>>>>>>>> Otherwise, how could a UTM get the right answer, since it only
>>>>>>>>>> gets
>>>>>>>>>> the description.
>>>>>>>>>
>>>>>>>>> We can get around all of this stuff by simply using this criteria:
>>>>>>>>> Date 10/13/2022 11:29:23 AM
>>>>>>>>> *MIT Professor Michael Sipser agreed this verbatim paragraph is
>>>>>>>>> correct*
>>>>>>>>> (He has neither reviewed nor agreed to anything else in this
>>>>>>>>> paper)
>>>>>>>>> (a) If simulating halt decider H correctly simulates its input D
>>>>>>>>> until H
>>>>>>>>> correctly determines that its simulated D would never stop running
>>>>>>>>> unless aborted then
>>>>>>>>> (b) H can abort its simulation of D and correctly report that D
>>>>>>>>> specifies a non-halting sequence of configurations.
>>>>>>>>>
>>>>>>>>> *When we apply this criteria* (elaborated above)
>>>>>>>>> Will you halt if you never abort your simulation?
>>>>>>>>> *Then the halting problem is conquered*
>>>>>>>>>
>>>>>>>>> When two different machines implementing this criteria
>>>>>>>>> get different results from identical inputs then we
>>>>>>>>> know that Pathological Self-Reference has been detected.
>>>>>>>>>
>>>>>>>>> We don't even need to know that for:
>>>>>>>>> *denial-of-service-attack detection*
>>>>>>>>> *NO always means reject as unsafe*
>>>>>>>>>
>>>>>>>>
>>>>>>>> But, Halting theorem never said "there's an input that halts
>>>>>>>> all machines", it just says "for any machine, there's an input
>>>>>>>> that halts it".
>>>>>>>>
>>>>>>>> Where "halt the machine" means "put it in an infinite loop".
>>>>>>>>
>>>>>>>> So, rather, Halting theorem never said "there's an input that
>>>>>>>> exhausts all machines", it just says, "for any machine, there's
>>>>>>>> an input that exhausts it".
>>>>>>>>
>>>>>>>> I still don't see how that would be with infinite tapes though,
>>>>>>>> without a means of checking all the way right the tape in one
>>>>>>>> step, i.e. that it's 1's or 0's or any pattern at all, any
>>>>>>>> input that unbounded with respect to the machine basically
>>>>>>>> exhausts it or where the machine would run in the unbounded.
>>>>>>>>
>>>>>>>>
>>>>>>>> Of course any finite tape, has a static analysis that is
>>>>>>>> not infinite, that decides whether or not it halts
>>>>>>>> (or, loops, or grows, the state space of the decider).
>>>>>>>>
>>>>>>>> Static analysis has to either enumerate or _infer_ the
>>>>>>>> state-space, where equal values in what's determined
>>>>>>>> the idempotent can detect loops, while inequalities
>>>>>>>> or proven divergence, ..., can detect unbounded growth.
>>>>>>>>
>>>>>>>> Now, proving convergence or divergence is its own kind
>>>>>>>> of thing. For example, there are series that converge
>>>>>>>> very slowly, and series that diverge very slowly. These
>>>>>>>> are subtly intractable to analysis.
>>>>>>>>
>>>>>>>> Then the usual idea of progressions that rapidly grow
>>>>>>>> yet return to what's detectable, are pathological to
>>>>>>>> analysis, only in resources not correctness or completion,
>>>>>>>> vis-a-vis the subtle intractability of the convergence or
>>>>>>>> divergence what either halts, or loops, or grows unboundedly.
>>>>>>>>
>>>>>>>> Separately "not-halts" into "loops or grows unboundedly",
>>>>>>>> has that really for most matters of interest of understanding
>>>>>>>> the state-space of programs, is "halts or enters loops"
>>>>>>>> and not "grows unboundedly".
>>>>>>>>
>>>>>>>> I.e. if the Halting problem is basically subject the
>>>>>>>> subtle intractability of slow convergence, that otherwise
>>>>>>>> it can just infer divergence and decide, practically
>>>>>>>> it's sort of more relevant what the machine would be
>>>>>>>> doing on the input on the tape, then with respect to
>>>>>>>> beyond the Turing theory, of the state of the read-head,
>>>>>>>> what happens when somebody modifies the tape, or events,
>>>>>>>> the write-head.
>>>>>>>>
>>>>>>>> Anyways though for bounded inputs, besides slow divergence,
>>>>>>>> it's to be made clear that _most all_ and _almost all_
>>>>>>>> programs _are_ decided their behavior by static analysis.
>>>>>>>>
>>>>>>>> Though, "most all" and "almost all" might be a bit strong,
>>>>>>>> but pretty much all that don't involve "the subtle intractability
>>>>>>>> of slow divergence".
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> Giving the idea that an existence result
>>>>>>>> is in any way the expected result here
>>>>>>>> seems sort of the root of this dilem-na.
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> (Though that the real numbers in ZFC have a well-ordering
>>>>>>>> and if they had a normal ordering that was a well-ordering,
>>>>>>>> that would be a thing, because ZFC has a well-ordering of
>>>>>>>> [0,1], but can't give one.)
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> What I'm saying is that you'd need to solve all
>>>>>>> cases of slow convergence and slow divergence
>>>>>>> to have a correct and complete halt decider,
>>>>>>> for the unbounded, and finite, if not the infinite.
>>>>>>>
>>>>>>> Most though setups of convergence though
>>>>>>> would effectively exhaust according to the
>>>>>>> existence of criteria of convergence as modeling
>>>>>>> the computation, though, so, if you exclude
>>>>>>> slow convergence and slow divergence,
>>>>>>> then a subset of very usual machines is of
>>>>>>> course quite all decidable.  (Or decide-able,
>>>>>>> or as I put it once, "not deciddable".)
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> Well-ordering the reals in ZFC, that's it own
>>>>>>> sort issue - that a well-ordering implies the
>>>>>>> existence of a bijection from ordinals to a
>>>>>>> set, then as that as tuples, subsets of those
>>>>>>> are well-ordered by their ordinal part, then
>>>>>>> that if ever uncountably many of those are
>>>>>>> in normal order their real part and irrationals,
>>>>>>> then between those are rationals.  I.e.,
>>>>>>> ZFC doesn't give an example of well-ordering
>>>>>>> the reals, but there is one.
>>>>>>>
>>>>>>>
>>>>>>> So, Peter, I think you're kind of misreading that
>>>>>>> quote you authoritated.  It just says "if static
>>>>>>> analysis detects a loop or unboundedly growing
>>>>>>> then it's not-halts".
>>>>>>>
>>>>>>>
>>>>>>> Of course, for any bounded resource, there
>>>>>>> are arbitrarily larger bounded resources
>>>>>>> required by what's called pathological,
>>>>>>> that an arbitrarily larger resource can though solve.
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> So anyways "slow convergence" and "slow divergence",
>>>>>>> have that in mathematics, it's unknown for some series
>>>>>>> whether they converge or diverge, though it's usually
>>>>>>> accepted that the inverse powers of 2 converge and
>>>>>>> that the sum of integers diverges, and that periodic
>>>>>>> functions do neither.
>>>>>>>
>>>>>>> I.e. the criteria of convergence and existence of a limit,
>>>>>>> and the special case "diverges as going to infinity",
>>>>>>> are not yet closed in today's mathematics.
>>>>>>>
>>>>>>> It's sort of a conjecture or independent whether
>>>>>>> they ever could be, closed, and complete, the criteria.
>>>>>>>
>>>>>>>
>>>>>>
>>>>>> I have spent 20 years on The conventional halting problem proofs,
>>>>>> Gödel 1931 Incompleteness and the Liar Paradox. I have only spent
>>>>>> about seven years on Tarski Undefinability.
>>>>>>
>>>>>> The Halting problem is the only one of these that has a formal system
>>>>>> that cannot have hidden gaps its its reasoning. If a machine can
>>>>>> do it
>>>>>> then it can be done.
>>>>>>
>>>>>
>>>>> "Algorithmics" is a study of algorithms. The most usual
>>>>> consideration is "asymptotics", "asymptotics of complexity",
>>>>> with the idea that computing does work, that there are
>>>>> closures and completions, and there are approximations and
>>>>> attenuations, as it were, in the full and the final and
>>>>> the initial and the partial.
>>>>>
>>>>> The most usual resources are time, and space, that actions
>>>>> occur in time, and according to a structure, in space.
>>>>>
>>>>> Of course any CS grad pretty much knows that.
>>>>>
>>>>>
>>>>> The objects of mathematics, have it so, that the finite
>>>>> and bounded resources of any collection of digital and
>>>>> binary machines, are finite and bounded, while, the unbounded
>>>>> models of mathematics that represent them, are unbounded,
>>>>> it's bad that anybody ever said a Turing tape was "infinite",
>>>>> except that it's unbounded as an object of mathematics, and
>>>>> the model of any finite, digital, binary machine.
>>>>>
>>>>> One might aver "digital" has a complement in computing,
>>>>> "analog", which represents the continuous instead of
>>>>> the discrete, and one might aver "binary" has a complement,
>>>>> either "real-valued" or "fuzzy" or variously about the
>>>>> "multi-valent", in logic, the models, according to the
>>>>> mechanisms, the models, their mechanics, machines.
>>>>>
>>>>> The Turing machine though is a model of unbounded machine
>>>>> as what any thusly simply mechanical model can implement,
>>>>> up to its bounds.
>>>>>
>>>>>
>>>>> So, the objects of mathematics, then involve analytical
>>>>> results. I can't draw a circle, though a compass may
>>>>> portray a diagram arbitrarily close, and no different.
>>>>> It's the analytical results, that make so for why
>>>>> mathematics is: "sensible, fungible, and tractable",
>>>>> and for "the ubiquitous success of mathematics" in
>>>>> all such matters of physics, which is a continuum mechanics,
>>>>> and as well discrete mechanics.
>>>>>
>>>>> Of course, results exist in mathematical objects,
>>>>> after the unbounded the unbounded and complete,
>>>>> which is why closures and completions are so
>>>>> relevant to all such matters, vis-a-vis approximations
>>>>> and attenuations, which in bounded terms result
>>>>> indistinguishably non-different results,
>>>>> up to precision, say, or tolerances.
>>>>>
>>>>> So, "machine" might be discrete, binary, and digital,
>>>>> or it might be continuous, multivalent, and analog.
>>>>>
>>>>> The Turing machine in a concrete form, is bounded.
>>>>>
>>>>> Similar models, here mostly for the stack models,
>>>>> stack machines, among finite-state-machines then
>>>>> for example concepts like "unbounded nondeterminism",
>>>>> here is pretty much about determinism, about the
>>>>> defined behavior of a machine according to a
>>>>> mathematical model that given defined behavior
>>>>> of the mechanisms results a mechanical model of
>>>>> a mathematical model, such a theory as this.
>>>>>
>>>>> So, the theory of computing, with mechanical or
>>>>> computational models, models of computing, is
>>>>> pretty much exactly like the theory of physics,
>>>>> with the attachment of physical models to the
>>>>> mathematical models, and rather, "interpreted as",
>>>>> the models, the models, with respect to each other.
>>>>>
>>>>> I.e. the extensionality that so follows "they're
>>>>> equal, or same, or not different, they represent
>>>>> each other, they're equi-interpretable, they model
>>>>> each other", the models, of logical and mathematical
>>>>> and computational and mechanical and physical models,
>>>>> help represent that over all the entire thing there
>>>>> is a usual theory for the entire thing, it's a theory
>>>>> where model theory models extensionality, and in identity
>>>>> intensionality, about equality, tautology, identity,
>>>>> and sameness/difference, and nearness/farness, and all
>>>>> these usual aspects of the mathematical models'
>>>>> arithmetic, algebra, and geometry. (And function
>>>>> theory and topology, with respect to categories and
>>>>> types, sets and parts, relations and predicates,
>>>>> then all the model-building among that which is
>>>>> all the propositional or the stipulated or axiomatic.)
>>>>>
>>>>> The idea is that this sort of platonistic universe of
>>>>> mathematical objects has always existed, and it's just
>>>>> discovered, then that axiomatics for example, just
>>>>> sort of results a model theory of it, with regards
>>>>> to models, the modular, modulation, modality, and the mode.
>>>>>
>>>>>
>>>>> So, a machine, with respect to computation, establishing
>>>>> the validity of interpretations of models, is subject to
>>>>> filling out all the above, vis-a-vis what it can do,
>>>>> and it can-do.
>>>>>
>>>>>
>>>>> Then, "slowly converging" and "slowly diverging"
>>>>> are examples that get get into that there are more
>>>>> law(s) of large numbers, than the usual classical
>>>>> law of large numbers.
>>>>>
>>>>> Some people variously do or don't have a mathematical
>>>>> model of larger numbers, or the infinite, at all.
>>>>> It's a totally ancient and dogmatic tradition that
>>>>> no, we finite peoples, persons, or agents of limited
>>>>> and finite means, no we do not have discrete, digital,
>>>>> binary mechanical models of the infinite.
>>>>>
>>>>> Yet, it's also about the first thing that deduction
>>>>> arrives at just from the plain contemplation of the
>>>>> beyond the unbounded, like the theories of Democritus
>>>>> and Atomism or the theories of Zeno and Continuity,
>>>>> that make it so we at least have something to arrive
>>>>> at for models of the unbounded, as "methods of exhaustion",
>>>>> the approximation and attenuative what results
>>>>> the asymptotics of algorithmics.
>>>>>
>>>>> So, we do have mental models of the continuous and infinite.
>>>>> And, where physics is a continuum mechanics,
>>>>> there are physical models of it as well, then
>>>>> with regards to the philosophy of science and
>>>>> the scientific method and the Big Science and
>>>>> from the Primary Science, lots to establish
>>>>> that the continuous and analog has mathematical
>>>>> results, as whethe, "sensible, fungible, and tractable",
>>>>> to machines at all.
>>>>>
>>>>> Numerical methods then, and, approximation and
>>>>> attenuation, result from analysis, why they exist,
>>>>> here for the criteria of convergence and existence
>>>>> of limits, in the closed, in the complete.
>>>>>
>>>>>
>>>>> So, a metonymy, unless it's "the Strong Metonymy",
>>>>> which is utter intensionality of "the ground model",
>>>>> is yet a metaphor and a metaphor yet a weaker metonymy
>>>>> joint among weaker metonymys, that, ontology, has
>>>>> that there are or may be a completion, of ontology,
>>>>> as a "strong ontology" and "strong metonymy",
>>>>> but otherwise it's just a bag of facts.
>>>>>
>>>>>
>>>>>
>>>>> Here then there's certainly a perceived analytical
>>>>> development Cantor space, and, Square Cantor space,
>>>>> and, Signal Cantor space as it were, about that among
>>>>> the law(s) of large numbers, there are definitions of
>>>>> continuity, at least including field-reals, line-reals,
>>>>> and signal-reals, three sorts continuous domains.
>>>>>
>>>>>
>>>>> Mathematics owes physics this to better begin to
>>>>> explain, to disclose, to find truths of, continuum mechanics.
>>>>>
>>>>>
>>>>> Then that analog, fuzzy, multi-value, "quantum" computing
>>>>> models (parts) of that, that discrete, binary, digital
>>>>> computing does not, is a thing.
>>>>>
>>>>>
>>>>>
>>>>> The convergence of terms in series is an example
>>>>> of a model of things, that, sometimes has clear
>>>>> and direct and perfectly accurate representations,
>>>>> models, in stack machine, and sometimes doesn't.
>>>>>
>>>>>
>>>>> Of course, for any bounded input, there is a sufficiently
>>>>> larger bounded analyzer, that does solve it. So, why not
>>>>> just put those all together? It would be infinite,
>>>>> yet most things are not. (Or, you know, are.)
>>>>>
>>>>>
>>>>> It's called "foundations" the study of all these things
>>>>> as a fundamental coherency, a thing together, while of
>>>>> its parts.
>>>>>
>>>>> So, students should well have the concepts of "abacus"
>>>>> and "slide rule" and other "tabulators and computers"
>>>>> of the usual manual mechanical variety down, according
>>>>> to the principles of the mathematical models behind them,
>>>>> then introducing the law(s) of large numbers, then
>>>>> introducing the idea of a standard model of an infinite,
>>>>> about rates, related rates, asymptotics, and bounds.
>>>>>
>>>>>
>>>>> "Foundations" then the idea is that it's been around
>>>>> forever, it's our dogma and from our canon and it develops
>>>>> over time, and the current edition is called "modern".
>>>>
>>>> Did I make any mistakes?
>>>>
>>>> Tarski anchors his proof in the Liar Paradox
>>>> https://liarparadox.org/Tarski_247_248.pdf
>>>>
>>>> How Tarski encodes the Liar Paradox
>>>> x ∉ True if and only if p
>>>> where the symbol 'p' represents the whole sentence x
>>>>
>>>> This is transformed into line 01 of his proof
>>>> by replacing True with Provable
>>>>
>>>> *Here is the Tarski Undefinability Theorem proof*
>>>> (1) x ∉ Provable if and only if p   // assumption
>>>> (2) x ∈ True if and only if p       // assumption
>>>> (3) x ∉ Provable if and only if x ∈ True.
>>>> (4) either x ∉ True or x̄ ∉ True;    // axiom: ~True(x) ∨ ~True(~x)
>>>> (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
>>>> (6) if x̄ ∈ Provable, then x̄ ∈ True;  // axiom: Provable(~x) → True(~x)
>>>> (7) x ∈ True
>>>> (8) x ∉ Provable
>>>> (9) x̄ ∉ Provable
>>>>
>>>> Tarski's Full proof
>>>> https://liarparadox.org/Tarski_275_276.pdf
>>>>
>>>>
>>>>
>>>
>>>
>>> Well I have some collected works of Tarski so
>>> I'll be looking to them, the excerpt there you
>>> reference basically talks about the model under
>>> consideration, as a fragment, of "the meta-theory",
>>> i.e., "Tarski's meta-theory" is "rather underdefined",
>>> that what he does is that he says that in the meta-theory,
>>> that there's something about the model under consideration,
>>> that isn't true in the model but is true, and he invokes
>>> Goedel to basically says that Goedel's incompleteness
>>> applies to this model, which of course must thusly by
>>> "strong enough to support arithmetic", where otherwise
>>> of course, Goedel's incompleteness does _not_ apply
>>> to theories that may be entirely closed categories.
>>>
>>> It seems the issue is that you think that you have
>>> entirely closed categories, in your meta-theory,
>>> then are applying this meta-theory to another model
>>> under consideration, which isn't, so it seems like
>>> you have implicits on your model, which are underdefined,
>>> rather as the "properly logical" or "non-logical" are
>>> with respect to being modeled by the logical, that
>>> you have the some sort of "un-logical" that are the
>>> result of that the axioms are un-logical, with regards
>>> to that your "meta-theory" is logical and your model
>>> under consideration, its _logical_ objects, are actually
>>> "un-logical" objects with respect to your, "meta-theory".
>>>
>>>
>>> That's a problem overall with "meta-theory", here the
>>> goal is to arrive at a theory that is its own meta-theory,
>>> that's also all one language, with regards to otherwise
>>
>> Yes that is exactly correct.
>>
>> *True(L,x) returns TRUE when x is True otherwise returns FALSE*
>
> So what does True(L, S) return when S is defined as S := ~True(L,S)
>

On 3/14/2024 11:58 AM, Ross Finlayson wrote:
> On 03/13/2024 10:20 PM, olcott wrote:
>> On 3/13/2024 1:16 PM, Ross Finlayson wrote:
>>> On 03/12/2024 09:00 PM, olcott wrote:
>>>> On 3/12/2024 10:49 PM, Ross Finlayson wrote:
>>>>> On 03/12/2024 08:23 PM, Ross Finlayson wrote:
>>>>>> On 03/12/2024 07:52 PM, olcott wrote:
>>>>>>> On 3/12/2024 9:28 PM, Richard Damon wrote:
>>>>>>>> On 3/12/24 4:31 PM, olcott wrote:
>>>>>>>>> On 3/12/2024 6:11 PM, Richard Damon wrote:
>>>>>>>>>> On 3/12/24 3:53 PM, olcott wrote:
>>>>>>>>>>> On 3/12/2024 5:30 PM, Richard Damon wrote:
>>>>>>>>>>>> On 3/12/24 2:34 PM, olcott wrote:
>>>>>>>>>>>>> On 3/12/2024 4:23 PM, Richard Damon wrote:
>>>>>>>>>>>>>> On 3/12/24 1:11 PM, olcott wrote:
>>>>>>>>>>>>>>> On 3/12/2024 2:40 PM, Richard Damon wrote:
>>>>>>>>>>>>>>>> On 3/12/24 12:02 PM, olcott wrote:
>>>>>>>>>>>>>>>>> On 3/12/2024 1:31 PM, immibis wrote:
>>>>>>>>>>>>>>>>>> On 12/03/24 19:12, olcott wrote:
>>>>>>>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> There is some input TMD to every H such that
>>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> And it can be a different TMD to each H.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> When we disallow decider/input pairs that are incorrect
>>>>>>>>>>>>>>>>>>> questions where both YES and NO are the wrong answer
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> Once we understand that either YES or NO is the right
>>>>>>>>>>>>>>>>>> answer, the whole rebuttal is tossed out as invalid and
>>>>>>>>>>>>>>>>>> incorrect.
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩
>>>>>>>>>>>>>>>>> halts
>>>>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn   // Ĥ applied to ⟨Ĥ⟩
>>>>>>>>>>>>>>>>> does
>>>>>>>>>>>>>>>>> not halt
>>>>>>>>>>>>>>>>> BOTH YES AND NO ARE THE WRONG ANSWER FOR EVERY Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> No, because a given H will only go to one of the answers.
>>>>>>>>>>>>>>>> THAT
>>>>>>>>>>>>>>>> will be wrong, and the other one right.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Not exactly. A pair of otherwise identical machines that
>>>>>>>>>>>>>>> (that are contained within the above specified set)
>>>>>>>>>>>>>>> only differ by return value will both be wrong on the
>>>>>>>>>>>>>>> same pathological input.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> You mean a pair of DIFFERENT machines. Any difference is
>>>>>>>>>>>>>> different.
>>>>>>>>>>>>>
>>>>>>>>>>>>> Every decider/input pair (referenced in the above set) has a
>>>>>>>>>>>>> corresponding decider/input pair that only differs by the
>>>>>>>>>>>>> return
>>>>>>>>>>>>> value of its decider.
>>>>>>>>>>>>
>>>>>>>>>>>> Nope.
>>>>>>>>>>>>
>>>>>>>>>>> ∀ H ∈ Turing_Machines_Returning_Boolean
>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>
>>>>>>>>>>> Every H/TMD pair (referenced in the above set) has a
>>>>>>>>>>> corresponding H/TMD pair that only differs by the return
>>>>>>>>>>> value of its Boolean_TM.
>>>>>>>>>>
>>>>>>>>>> That isn't in the set above.
>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> That both of these H/TMD pairs get the wrong answer proves that
>>>>>>>>>>> their question was incorrect because the opposite answer to the
>>>>>>>>>>> same question is also proven to be incorrect.
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>> Nope, since both aren't in the set selected.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> When they are deciders that must get the correct answer both
>>>>>>>>> of them are not in the set.
>>>>>>>>
>>>>>>>> *IF* they are correct decider.
>>>>>>>>
>>>>>>>> WHen we select from all Turing Machine Deciders, there is no
>>>>>>>> requirement that any of them get any particular answer right.
>>>>>>>>
>>>>>>>> So, ALL deciders are in the set that we cycle through and apply the
>>>>>>>> following logic to ALL of them.
>>>>>>>>
>>>>>>>> Each is them paired with an input that it will get wrong, and the
>>>>>>>> existance of the input was what as just proven, the ^ template
>>>>>>>>
>>>>>>>>>
>>>>>>>>> When they are Turing_Machines_Returning_Boolean the this
>>>>>>>>> set inherently includes identical pairs that only differ
>>>>>>>>> by return value.
>>>>>>>>
>>>>>>>> But in the step of select and input that they will get wrong, they
>>>>>>>> will be givne DIFFERENT inputs.
>>>>>>>>
>>>>>>>>>
>>>>>>>>>> You just don't understand what that statement is saying.
>>>>>>>>>>
>>>>>>>>>> I've expalined it, but it seems over you head.
>>>>>>>>>>
>>>>>>>>> No the problem is that you are not paying attention.
>>>>>>>>
>>>>>>>> No, you keep on making STUPID mistakes, like thinking that select a
>>>>>>>> input that the machine will get wrong needs to be the same for two
>>>>>>>> differnt machines.
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>>>
>>>>>>>>>> For Every H, we show we can find at least one input (chosen
>>>>>>>>>> just for
>>>>>>>>>> that machine) that it will get wrong.
>>>>>>>>>>
>>>>>>>>> When we use machine templates then we can see instances of
>>>>>>>>> the same machine that only differs by return value where both
>>>>>>>>> get the wrong answer on the same input. By same input I mean
>>>>>>>>> the same finite string of numerical values.
>>>>>>>>>
>>>>>>>>
>>>>>>>> But if they returned differnt values, they will have different
>>>>>>>> descriptions.
>>>>>>>>
>>>>>>>> Otherwise, how could a UTM get the right answer, since it only gets
>>>>>>>> the description.
>>>>>>>
>>>>>>> We can get around all of this stuff by simply using this criteria:
>>>>>>> Date 10/13/2022 11:29:23 AM
>>>>>>> *MIT Professor Michael Sipser agreed this verbatim paragraph is
>>>>>>> correct*
>>>>>>> (He has neither reviewed nor agreed to anything else in this paper)
>>>>>>> (a) If simulating halt decider H correctly simulates its input D
>>>>>>> until H
>>>>>>> correctly determines that its simulated D would never stop running
>>>>>>> unless aborted then
>>>>>>> (b) H can abort its simulation of D and correctly report that D
>>>>>>> specifies a non-halting sequence of configurations.
>>>>>>>
>>>>>>> *When we apply this criteria* (elaborated above)
>>>>>>> Will you halt if you never abort your simulation?
>>>>>>> *Then the halting problem is conquered*
>>>>>>>
>>>>>>> When two different machines implementing this criteria
>>>>>>> get different results from identical inputs then we
>>>>>>> know that Pathological Self-Reference has been detected.
>>>>>>>
>>>>>>> We don't even need to know that for:
>>>>>>> *denial-of-service-attack detection*
>>>>>>> *NO always means reject as unsafe*
>>>>>>>
>>>>>>
>>>>>> But, Halting theorem never said "there's an input that halts
>>>>>> all machines", it just says "for any machine, there's an input
>>>>>> that halts it".
>>>>>>
>>>>>> Where "halt the machine" means "put it in an infinite loop".
>>>>>>
>>>>>> So, rather, Halting theorem never said "there's an input that
>>>>>> exhausts all machines", it just says, "for any machine, there's
>>>>>> an input that exhausts it".
>>>>>>
>>>>>> I still don't see how that would be with infinite tapes though,
>>>>>> without a means of checking all the way right the tape in one
>>>>>> step, i.e. that it's 1's or 0's or any pattern at all, any
>>>>>> input that unbounded with respect to the machine basically
>>>>>> exhausts it or where the machine would run in the unbounded.
>>>>>>
>>>>>>
>>>>>> Of course any finite tape, has a static analysis that is
>>>>>> not infinite, that decides whether or not it halts
>>>>>> (or, loops, or grows, the state space of the decider).
>>>>>>
>>>>>> Static analysis has to either enumerate or _infer_ the
>>>>>> state-space, where equal values in what's determined
>>>>>> the idempotent can detect loops, while inequalities
>>>>>> or proven divergence, ..., can detect unbounded growth.
>>>>>>
>>>>>> Now, proving convergence or divergence is its own kind
>>>>>> of thing. For example, there are series that converge
>>>>>> very slowly, and series that diverge very slowly. These
>>>>>> are subtly intractable to analysis.
>>>>>>
>>>>>> Then the usual idea of progressions that rapidly grow
>>>>>> yet return to what's detectable, are pathological to
>>>>>> analysis, only in resources not correctness or completion,
>>>>>> vis-a-vis the subtle intractability of the convergence or
>>>>>> divergence what either halts, or loops, or grows unboundedly.
>>>>>>
>>>>>> Separately "not-halts" into "loops or grows unboundedly",
>>>>>> has that really for most matters of interest of understanding
>>>>>> the state-space of programs, is "halts or enters loops"
>>>>>> and not "grows unboundedly".
>>>>>>
>>>>>> I.e. if the Halting problem is basically subject the
>>>>>> subtle intractability of slow convergence, that otherwise
>>>>>> it can just infer divergence and decide, practically
>>>>>> it's sort of more relevant what the machine would be
>>>>>> doing on the input on the tape, then with respect to
>>>>>> beyond the Turing theory, of the state of the read-head,
>>>>>> what happens when somebody modifies the tape, or events,
>>>>>> the write-head.
>>>>>>
>>>>>> Anyways though for bounded inputs, besides slow divergence,
>>>>>> it's to be made clear that _most all_ and _almost all_
>>>>>> programs _are_ decided their behavior by static analysis.
>>>>>>
>>>>>> Though, "most all" and "almost all" might be a bit strong,
>>>>>> but pretty much all that don't involve "the subtle intractability
>>>>>> of slow divergence".
>>>>>>
>>>>>>
>>>>>>
>>>>>> Giving the idea that an existence result
>>>>>> is in any way the expected result here
>>>>>> seems sort of the root of this dilem-na.
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> (Though that the real numbers in ZFC have a well-ordering
>>>>>> and if they had a normal ordering that was a well-ordering,
>>>>>> that would be a thing, because ZFC has a well-ordering of
>>>>>> [0,1], but can't give one.)
>>>>>>
>>>>>>
>>>>>
>>>>>
>>>>>
>>>>> What I'm saying is that you'd need to solve all
>>>>> cases of slow convergence and slow divergence
>>>>> to have a correct and complete halt decider,
>>>>> for the unbounded, and finite, if not the infinite.
>>>>>
>>>>> Most though setups of convergence though
>>>>> would effectively exhaust according to the
>>>>> existence of criteria of convergence as modeling
>>>>> the computation, though, so, if you exclude
>>>>> slow convergence and slow divergence,
>>>>> then a subset of very usual machines is of
>>>>> course quite all decidable.  (Or decide-able,
>>>>> or as I put it once, "not deciddable".)
>>>>>
>>>>>
>>>>>
>>>>> Well-ordering the reals in ZFC, that's it own
>>>>> sort issue - that a well-ordering implies the
>>>>> existence of a bijection from ordinals to a
>>>>> set, then as that as tuples, subsets of those
>>>>> are well-ordered by their ordinal part, then
>>>>> that if ever uncountably many of those are
>>>>> in normal order their real part and irrationals,
>>>>> then between those are rationals.  I.e.,
>>>>> ZFC doesn't give an example of well-ordering
>>>>> the reals, but there is one.
>>>>>
>>>>>
>>>>> So, Peter, I think you're kind of misreading that
>>>>> quote you authoritated.  It just says "if static
>>>>> analysis detects a loop or unboundedly growing
>>>>> then it's not-halts".
>>>>>
>>>>>
>>>>> Of course, for any bounded resource, there
>>>>> are arbitrarily larger bounded resources
>>>>> required by what's called pathological,
>>>>> that an arbitrarily larger resource can though solve.
>>>>>
>>>>>
>>>>>
>>>>> So anyways "slow convergence" and "slow divergence",
>>>>> have that in mathematics, it's unknown for some series
>>>>> whether they converge or diverge, though it's usually
>>>>> accepted that the inverse powers of 2 converge and
>>>>> that the sum of integers diverges, and that periodic
>>>>> functions do neither.
>>>>>
>>>>> I.e. the criteria of convergence and existence of a limit,
>>>>> and the special case "diverges as going to infinity",
>>>>> are not yet closed in today's mathematics.
>>>>>
>>>>> It's sort of a conjecture or independent whether
>>>>> they ever could be, closed, and complete, the criteria.
>>>>>
>>>>>
>>>>
>>>> I have spent 20 years on The conventional halting problem proofs,
>>>> Gödel 1931 Incompleteness and the Liar Paradox. I have only spent
>>>> about seven years on Tarski Undefinability.
>>>>
>>>> The Halting problem is the only one of these that has a formal system
>>>> that cannot have hidden gaps its its reasoning. If a machine can do it
>>>> then it can be done.
>>>>
>>>
>>> "Algorithmics" is a study of algorithms. The most usual
>>> consideration is "asymptotics", "asymptotics of complexity",
>>> with the idea that computing does work, that there are
>>> closures and completions, and there are approximations and
>>> attenuations, as it were, in the full and the final and
>>> the initial and the partial.
>>>
>>> The most usual resources are time, and space, that actions
>>> occur in time, and according to a structure, in space.
>>>
>>> Of course any CS grad pretty much knows that.
>>>
>>>
>>> The objects of mathematics, have it so, that the finite
>>> and bounded resources of any collection of digital and
>>> binary machines, are finite and bounded, while, the unbounded
>>> models of mathematics that represent them, are unbounded,
>>> it's bad that anybody ever said a Turing tape was "infinite",
>>> except that it's unbounded as an object of mathematics, and
>>> the model of any finite, digital, binary machine.
>>>
>>> One might aver "digital" has a complement in computing,
>>> "analog", which represents the continuous instead of
>>> the discrete, and one might aver "binary" has a complement,
>>> either "real-valued" or "fuzzy" or variously about the
>>> "multi-valent", in logic, the models, according to the
>>> mechanisms, the models, their mechanics, machines.
>>>
>>> The Turing machine though is a model of unbounded machine
>>> as what any thusly simply mechanical model can implement,
>>> up to its bounds.
>>>
>>>
>>> So, the objects of mathematics, then involve analytical
>>> results. I can't draw a circle, though a compass may
>>> portray a diagram arbitrarily close, and no different.
>>> It's the analytical results, that make so for why
>>> mathematics is: "sensible, fungible, and tractable",
>>> and for "the ubiquitous success of mathematics" in
>>> all such matters of physics, which is a continuum mechanics,
>>> and as well discrete mechanics.
>>>
>>> Of course, results exist in mathematical objects,
>>> after the unbounded the unbounded and complete,
>>> which is why closures and completions are so
>>> relevant to all such matters, vis-a-vis approximations
>>> and attenuations, which in bounded terms result
>>> indistinguishably non-different results,
>>> up to precision, say, or tolerances.
>>>
>>> So, "machine" might be discrete, binary, and digital,
>>> or it might be continuous, multivalent, and analog.
>>>
>>> The Turing machine in a concrete form, is bounded.
>>>
>>> Similar models, here mostly for the stack models,
>>> stack machines, among finite-state-machines then
>>> for example concepts like "unbounded nondeterminism",
>>> here is pretty much about determinism, about the
>>> defined behavior of a machine according to a
>>> mathematical model that given defined behavior
>>> of the mechanisms results a mechanical model of
>>> a mathematical model, such a theory as this.
>>>
>>> So, the theory of computing, with mechanical or
>>> computational models, models of computing, is
>>> pretty much exactly like the theory of physics,
>>> with the attachment of physical models to the
>>> mathematical models, and rather, "interpreted as",
>>> the models, the models, with respect to each other.
>>>
>>> I.e. the extensionality that so follows "they're
>>> equal, or same, or not different, they represent
>>> each other, they're equi-interpretable, they model
>>> each other", the models, of logical and mathematical
>>> and computational and mechanical and physical models,
>>> help represent that over all the entire thing there
>>> is a usual theory for the entire thing, it's a theory
>>> where model theory models extensionality, and in identity
>>> intensionality, about equality, tautology, identity,
>>> and sameness/difference, and nearness/farness, and all
>>> these usual aspects of the mathematical models'
>>> arithmetic, algebra, and geometry. (And function
>>> theory and topology, with respect to categories and
>>> types, sets and parts, relations and predicates,
>>> then all the model-building among that which is
>>> all the propositional or the stipulated or axiomatic.)
>>>
>>> The idea is that this sort of platonistic universe of
>>> mathematical objects has always existed, and it's just
>>> discovered, then that axiomatics for example, just
>>> sort of results a model theory of it, with regards
>>> to models, the modular, modulation, modality, and the mode.
>>>
>>>
>>> So, a machine, with respect to computation, establishing
>>> the validity of interpretations of models, is subject to
>>> filling out all the above, vis-a-vis what it can do,
>>> and it can-do.
>>>
>>>
>>> Then, "slowly converging" and "slowly diverging"
>>> are examples that get get into that there are more
>>> law(s) of large numbers, than the usual classical
>>> law of large numbers.
>>>
>>> Some people variously do or don't have a mathematical
>>> model of larger numbers, or the infinite, at all.
>>> It's a totally ancient and dogmatic tradition that
>>> no, we finite peoples, persons, or agents of limited
>>> and finite means, no we do not have discrete, digital,
>>> binary mechanical models of the infinite.
>>>
>>> Yet, it's also about the first thing that deduction
>>> arrives at just from the plain contemplation of the
>>> beyond the unbounded, like the theories of Democritus
>>> and Atomism or the theories of Zeno and Continuity,
>>> that make it so we at least have something to arrive
>>> at for models of the unbounded, as "methods of exhaustion",
>>> the approximation and attenuative what results
>>> the asymptotics of algorithmics.
>>>
>>> So, we do have mental models of the continuous and infinite.
>>> And, where physics is a continuum mechanics,
>>> there are physical models of it as well, then
>>> with regards to the philosophy of science and
>>> the scientific method and the Big Science and
>>> from the Primary Science, lots to establish
>>> that the continuous and analog has mathematical
>>> results, as whethe, "sensible, fungible, and tractable",
>>> to machines at all.
>>>
>>> Numerical methods then, and, approximation and
>>> attenuation, result from analysis, why they exist,
>>> here for the criteria of convergence and existence
>>> of limits, in the closed, in the complete.
>>>
>>>
>>> So, a metonymy, unless it's "the Strong Metonymy",
>>> which is utter intensionality of "the ground model",
>>> is yet a metaphor and a metaphor yet a weaker metonymy
>>> joint among weaker metonymys, that, ontology, has
>>> that there are or may be a completion, of ontology,
>>> as a "strong ontology" and "strong metonymy",
>>> but otherwise it's just a bag of facts.
>>>
>>>
>>>
>>> Here then there's certainly a perceived analytical
>>> development Cantor space, and, Square Cantor space,
>>> and, Signal Cantor space as it were, about that among
>>> the law(s) of large numbers, there are definitions of
>>> continuity, at least including field-reals, line-reals,
>>> and signal-reals, three sorts continuous domains.
>>>
>>>
>>> Mathematics owes physics this to better begin to
>>> explain, to disclose, to find truths of, continuum mechanics.
>>>
>>>
>>> Then that analog, fuzzy, multi-value, "quantum" computing
>>> models (parts) of that, that discrete, binary, digital
>>> computing does not, is a thing.
>>>
>>>
>>>
>>> The convergence of terms in series is an example
>>> of a model of things, that, sometimes has clear
>>> and direct and perfectly accurate representations,
>>> models, in stack machine, and sometimes doesn't.
>>>
>>>
>>> Of course, for any bounded input, there is a sufficiently
>>> larger bounded analyzer, that does solve it. So, why not
>>> just put those all together? It would be infinite,
>>> yet most things are not. (Or, you know, are.)
>>>
>>>
>>> It's called "foundations" the study of all these things
>>> as a fundamental coherency, a thing together, while of
>>> its parts.
>>>
>>> So, students should well have the concepts of "abacus"
>>> and "slide rule" and other "tabulators and computers"
>>> of the usual manual mechanical variety down, according
>>> to the principles of the mathematical models behind them,
>>> then introducing the law(s) of large numbers, then
>>> introducing the idea of a standard model of an infinite,
>>> about rates, related rates, asymptotics, and bounds.
>>>
>>>
>>> "Foundations" then the idea is that it's been around
>>> forever, it's our dogma and from our canon and it develops
>>> over time, and the current edition is called "modern".
>>
>> Did I make any mistakes?
>>
>> Tarski anchors his proof in the Liar Paradox
>> https://liarparadox.org/Tarski_247_248.pdf
>>
>> How Tarski encodes the Liar Paradox
>> x ∉ True if and only if p
>> where the symbol 'p' represents the whole sentence x
>>
>> This is transformed into line 01 of his proof
>> by replacing True with Provable
>>
>> *Here is the Tarski Undefinability Theorem proof*
>> (1) x ∉ Provable if and only if p   // assumption
>> (2) x ∈ True if and only if p       // assumption
>> (3) x ∉ Provable if and only if x ∈ True.
>> (4) either x ∉ True or x̄ ∉ True;    // axiom: ~True(x) ∨ ~True(~x)
>> (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
>> (6) if x̄ ∈ Provable, then x̄ ∈ True;  // axiom: Provable(~x) → True(~x)
>> (7) x ∈ True
>> (8) x ∉ Provable
>> (9) x̄ ∉ Provable
>>
>> Tarski's Full proof
>> https://liarparadox.org/Tarski_275_276.pdf
>>
>>
>>
>
>
> Well I have some collected works of Tarski so
> I'll be looking to them, the excerpt there you
> reference basically talks about the model under
> consideration, as a fragment, of "the meta-theory",
> i.e., "Tarski's meta-theory" is "rather underdefined",
> that what he does is that he says that in the meta-theory,
> that there's something about the model under consideration,
> that isn't true in the model but is true, and he invokes
> Goedel to basically says that Goedel's incompleteness
> applies to this model, which of course must thusly by
> "strong enough to support arithmetic", where otherwise
> of course, Goedel's incompleteness does _not_ apply
> to theories that may be entirely closed categories.
>
> It seems the issue is that you think that you have
> entirely closed categories, in your meta-theory,
> then are applying this meta-theory to another model
> under consideration, which isn't, so it seems like
> you have implicits on your model, which are underdefined,
> rather as the "properly logical" or "non-logical" are
> with respect to being modeled by the logical, that
> you have the some sort of "un-logical" that are the
> result of that the axioms are un-logical, with regards
> to that your "meta-theory" is logical and your model
> under consideration, its _logical_ objects, are actually
> "un-logical" objects with respect to your, "meta-theory".
>
>
> That's a problem overall with "meta-theory", here the
> goal is to arrive at a theory that is its own meta-theory,
> that's also all one language,

On 3/14/24 3:54 PM, olcott wrote:
> On 3/14/2024 5:01 PM, Richard Damon wrote:
>> On 3/14/24 12:01 PM, olcott wrote:
>>> On 3/14/2024 11:58 AM, Ross Finlayson wrote:
>>>> On 03/13/2024 10:20 PM, olcott wrote:
>>>>> On 3/13/2024 1:16 PM, Ross Finlayson wrote:
>>>>>> On 03/12/2024 09:00 PM, olcott wrote:
>>>>>>> On 3/12/2024 10:49 PM, Ross Finlayson wrote:
>>>>>>>> On 03/12/2024 08:23 PM, Ross Finlayson wrote:
>>>>>>>>> On 03/12/2024 07:52 PM, olcott wrote:
>>>>>>>>>> On 3/12/2024 9:28 PM, Richard Damon wrote:
>>>>>>>>>>> On 3/12/24 4:31 PM, olcott wrote:
>>>>>>>>>>>> On 3/12/2024 6:11 PM, Richard Damon wrote:
>>>>>>>>>>>>> On 3/12/24 3:53 PM, olcott wrote:
>>>>>>>>>>>>>> On 3/12/2024 5:30 PM, Richard Damon wrote:
>>>>>>>>>>>>>>> On 3/12/24 2:34 PM, olcott wrote:
>>>>>>>>>>>>>>>> On 3/12/2024 4:23 PM, Richard Damon wrote:
>>>>>>>>>>>>>>>>> On 3/12/24 1:11 PM, olcott wrote:
>>>>>>>>>>>>>>>>>> On 3/12/2024 2:40 PM, Richard Damon wrote:
>>>>>>>>>>>>>>>>>>> On 3/12/24 12:02 PM, olcott wrote:
>>>>>>>>>>>>>>>>>>>> On 3/12/2024 1:31 PM, immibis wrote:
>>>>>>>>>>>>>>>>>>>>> On 12/03/24 19:12, olcott wrote:
>>>>>>>>>>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> There is some input TMD to every H such that
>>>>>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> And it can be a different TMD to each H.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>>> When we disallow decider/input pairs that are
>>>>>>>>>>>>>>>>>>>>>> incorrect
>>>>>>>>>>>>>>>>>>>>>> questions where both YES and NO are the wrong answer
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>> Once we understand that either YES or NO is the right
>>>>>>>>>>>>>>>>>>>>> answer, the whole rebuttal is tossed out as invalid
>>>>>>>>>>>>>>>>>>>>> and
>>>>>>>>>>>>>>>>>>>>> incorrect.
>>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to
>>>>>>>>>>>>>>>>>>>> ⟨Ĥ⟩
>>>>>>>>>>>>>>>>>>>> halts
>>>>>>>>>>>>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn   // Ĥ applied to
>>>>>>>>>>>>>>>>>>>> ⟨Ĥ⟩
>>>>>>>>>>>>>>>>>>>> does
>>>>>>>>>>>>>>>>>>>> not halt
>>>>>>>>>>>>>>>>>>>> BOTH YES AND NO ARE THE WRONG ANSWER FOR EVERY Ĥ.H
>>>>>>>>>>>>>>>>>>>> ⟨Ĥ⟩ ⟨Ĥ⟩
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> No, because a given H will only go to one of the
>>>>>>>>>>>>>>>>>>> answers.
>>>>>>>>>>>>>>>>>>> THAT
>>>>>>>>>>>>>>>>>>> will be wrong, and the other one right.
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> ∀ H ∈ Turing_Machine_Deciders
>>>>>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> Not exactly. A pair of otherwise identical machines that
>>>>>>>>>>>>>>>>>> (that are contained within the above specified set)
>>>>>>>>>>>>>>>>>> only differ by return value will both be wrong on the
>>>>>>>>>>>>>>>>>> same pathological input.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> You mean a pair of DIFFERENT machines. Any difference is
>>>>>>>>>>>>>>>>> different.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> Every decider/input pair (referenced in the above set)
>>>>>>>>>>>>>>>> has a
>>>>>>>>>>>>>>>> corresponding decider/input pair that only differs by the
>>>>>>>>>>>>>>>> return
>>>>>>>>>>>>>>>> value of its decider.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> Nope.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>> ∀ H ∈ Turing_Machines_Returning_Boolean
>>>>>>>>>>>>>> ∃ TMD ∈ Turing_Machine_Descriptions  |
>>>>>>>>>>>>>> Predicted_Behavior(H, TMD) != Actual_Behavior(TMD)
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Every H/TMD pair (referenced in the above set) has a
>>>>>>>>>>>>>> corresponding H/TMD pair that only differs by the return
>>>>>>>>>>>>>> value of its Boolean_TM.
>>>>>>>>>>>>>
>>>>>>>>>>>>> That isn't in the set above.
>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> That both of these H/TMD pairs get the wrong answer proves
>>>>>>>>>>>>>> that
>>>>>>>>>>>>>> their question was incorrect because the opposite answer
>>>>>>>>>>>>>> to the
>>>>>>>>>>>>>> same question is also proven to be incorrect.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>> Nope, since both aren't in the set selected.
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> When they are deciders that must get the correct answer both
>>>>>>>>>>>> of them are not in the set.
>>>>>>>>>>>
>>>>>>>>>>> *IF* they are correct decider.
>>>>>>>>>>>
>>>>>>>>>>> WHen we select from all Turing Machine Deciders, there is no
>>>>>>>>>>> requirement that any of them get any particular answer right.
>>>>>>>>>>>
>>>>>>>>>>> So, ALL deciders are in the set that we cycle through and
>>>>>>>>>>> apply the
>>>>>>>>>>> following logic to ALL of them.
>>>>>>>>>>>
>>>>>>>>>>> Each is them paired with an input that it will get wrong, and
>>>>>>>>>>> the
>>>>>>>>>>> existance of the input was what as just proven, the ^ template
>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> When they are Turing_Machines_Returning_Boolean the this
>>>>>>>>>>>> set inherently includes identical pairs that only differ
>>>>>>>>>>>> by return value.
>>>>>>>>>>>
>>>>>>>>>>> But in the step of select and input that they will get wrong,
>>>>>>>>>>> they
>>>>>>>>>>> will be givne DIFFERENT inputs.
>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>> You just don't understand what that statement is saying.
>>>>>>>>>>>>>
>>>>>>>>>>>>> I've expalined it, but it seems over you head.
>>>>>>>>>>>>>
>>>>>>>>>>>> No the problem is that you are not paying attention.
>>>>>>>>>>>
>>>>>>>>>>> No, you keep on making STUPID mistakes, like thinking that
>>>>>>>>>>> select a
>>>>>>>>>>> input that the machine will get wrong needs to be the same
>>>>>>>>>>> for two
>>>>>>>>>>> differnt machines.
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>>> For Every H, we show we can find at least one input (chosen
>>>>>>>>>>>>> just for
>>>>>>>>>>>>> that machine) that it will get wrong.
>>>>>>>>>>>>>
>>>>>>>>>>>> When we use machine templates then we can see instances of
>>>>>>>>>>>> the same machine that only differs by return value where both
>>>>>>>>>>>> get the wrong answer on the same input. By same input I mean
>>>>>>>>>>>> the same finite string of numerical values.
>>>>>>>>>>>>
>>>>>>>>>>>
>>>>>>>>>>> But if they returned differnt values, they will have different
>>>>>>>>>>> descriptions.
>>>>>>>>>>>
>>>>>>>>>>> Otherwise, how could a UTM get the right answer, since it
>>>>>>>>>>> only gets
>>>>>>>>>>> the description.
>>>>>>>>>>
>>>>>>>>>> We can get around all of this stuff by simply using this
>>>>>>>>>> criteria:
>>>>>>>>>> Date 10/13/2022 11:29:23 AM
>>>>>>>>>> *MIT Professor Michael Sipser agreed this verbatim paragraph is
>>>>>>>>>> correct*
>>>>>>>>>> (He has neither reviewed nor agreed to anything else in this
>>>>>>>>>> paper)
>>>>>>>>>> (a) If simulating halt decider H correctly simulates its input D
>>>>>>>>>> until H
>>>>>>>>>> correctly determines that its simulated D would never stop
>>>>>>>>>> running
>>>>>>>>>> unless aborted then
>>>>>>>>>> (b) H can abort its simulation of D and correctly report that D
>>>>>>>>>> specifies a non-halting sequence of configurations.
>>>>>>>>>>
>>>>>>>>>> *When we apply this criteria* (elaborated above)
>>>>>>>>>> Will you halt if you never abort your simulation?
>>>>>>>>>> *Then the halting problem is conquered*
>>>>>>>>>>
>>>>>>>>>> When two different machines implementing this criteria
>>>>>>>>>> get different results from identical inputs then we
>>>>>>>>>> know that Pathological Self-Reference has been detected.
>>>>>>>>>>
>>>>>>>>>> We don't even need to know that for:
>>>>>>>>>> *denial-of-service-attack detection*
>>>>>>>>>> *NO always means reject as unsafe*
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> But, Halting theorem never said "there's an input that halts
>>>>>>>>> all machines", it just says "for any machine, there's an input
>>>>>>>>> that halts it".
>>>>>>>>>
>>>>>>>>> Where "halt the machine" means "put it in an infinite loop".
>>>>>>>>>
>>>>>>>>> So, rather, Halting theorem never said "there's an input that
>>>>>>>>> exhausts all machines", it just says, "for any machine, there's
>>>>>>>>> an input that exhausts it".
>>>>>>>>>
>>>>>>>>> I still don't see how that would be with infinite tapes though,
>>>>>>>>> without a means of checking all the way right the tape in one
>>>>>>>>> step, i.e. that it's 1's or 0's or any pattern at all, any
>>>>>>>>> input that unbounded with respect to the machine basically
>>>>>>>>> exhausts it or where the machine would run in the unbounded.
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Of course any finite tape, has a static analysis that is
>>>>>>>>> not infinite, that decides whether or not it halts
>>>>>>>>> (or, loops, or grows, the state space of the decider).
>>>>>>>>>
>>>>>>>>> Static analysis has to either enumerate or _infer_ the
>>>>>>>>> state-space, where equal values in what's determined
>>>>>>>>> the idempotent can detect loops, while inequalities
>>>>>>>>> or proven divergence, ..., can detect unbounded growth.
>>>>>>>>>
>>>>>>>>> Now, proving convergence or divergence is its own kind
>>>>>>>>> of thing. For example, there are series that converge
>>>>>>>>> very slowly, and series that diverge very slowly. These
>>>>>>>>> are subtly intractable to analysis.
>>>>>>>>>
>>>>>>>>> Then the usual idea of progressions that rapidly grow
>>>>>>>>> yet return to what's detectable, are pathological to
>>>>>>>>> analysis, only in resources not correctness or completion,
>>>>>>>>> vis-a-vis the subtle intractability of the convergence or
>>>>>>>>> divergence what either halts, or loops, or grows unboundedly.
>>>>>>>>>
>>>>>>>>> Separately "not-halts" into "loops or grows unboundedly",
>>>>>>>>> has that really for most matters of interest of understanding
>>>>>>>>> the state-space of programs, is "halts or enters loops"
>>>>>>>>> and not "grows unboundedly".
>>>>>>>>>
>>>>>>>>> I.e. if the Halting problem is basically subject the
>>>>>>>>> subtle intractability of slow convergence, that otherwise
>>>>>>>>> it can just infer divergence and decide, practically
>>>>>>>>> it's sort of more relevant what the machine would be
>>>>>>>>> doing on the input on the tape, then with respect to
>>>>>>>>> beyond the Turing theory, of the state of the read-head,
>>>>>>>>> what happens when somebody modifies the tape, or events,
>>>>>>>>> the write-head.
>>>>>>>>>
>>>>>>>>> Anyways though for bounded inputs, besides slow divergence,
>>>>>>>>> it's to be made clear that _most all_ and _almost all_
>>>>>>>>> programs _are_ decided their behavior by static analysis.
>>>>>>>>>
>>>>>>>>> Though, "most all" and "almost all" might be a bit strong,
>>>>>>>>> but pretty much all that don't involve "the subtle intractability
>>>>>>>>> of slow divergence".
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> Giving the idea that an existence result
>>>>>>>>> is in any way the expected result here
>>>>>>>>> seems sort of the root of this dilem-na.
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>>
>>>>>>>>> (Though that the real numbers in ZFC have a well-ordering
>>>>>>>>> and if they had a normal ordering that was a well-ordering,
>>>>>>>>> that would be a thing, because ZFC has a well-ordering of
>>>>>>>>> [0,1], but can't give one.)
>>>>>>>>>
>>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> What I'm saying is that you'd need to solve all
>>>>>>>> cases of slow convergence and slow divergence
>>>>>>>> to have a correct and complete halt decider,
>>>>>>>> for the unbounded, and finite, if not the infinite.
>>>>>>>>
>>>>>>>> Most though setups of convergence though
>>>>>>>> would effectively exhaust according to the
>>>>>>>> existence of criteria of convergence as modeling
>>>>>>>> the computation, though, so, if you exclude
>>>>>>>> slow convergence and slow divergence,
>>>>>>>> then a subset of very usual machines is of
>>>>>>>> course quite all decidable.  (Or decide-able,
>>>>>>>> or as I put it once, "not deciddable".)
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> Well-ordering the reals in ZFC, that's it own
>>>>>>>> sort issue - that a well-ordering implies the
>>>>>>>> existence of a bijection from ordinals to a
>>>>>>>> set, then as that as tuples, subsets of those
>>>>>>>> are well-ordered by their ordinal part, then
>>>>>>>> that if ever uncountably many of those are
>>>>>>>> in normal order their real part and irrationals,
>>>>>>>> then between those are rationals.  I.e.,
>>>>>>>> ZFC doesn't give an example of well-ordering
>>>>>>>> the reals, but there is one.
>>>>>>>>
>>>>>>>>
>>>>>>>> So, Peter, I think you're kind of misreading that
>>>>>>>> quote you authoritated.  It just says "if static
>>>>>>>> analysis detects a loop or unboundedly growing
>>>>>>>> then it's not-halts".
>>>>>>>>
>>>>>>>>
>>>>>>>> Of course, for any bounded resource, there
>>>>>>>> are arbitrarily larger bounded resources
>>>>>>>> required by what's called pathological,
>>>>>>>> that an arbitrarily larger resource can though solve.
>>>>>>>>
>>>>>>>>
>>>>>>>>
>>>>>>>> So anyways "slow convergence" and "slow divergence",
>>>>>>>> have that in mathematics, it's unknown for some series
>>>>>>>> whether they converge or diverge, though it's usually
>>>>>>>> accepted that the inverse powers of 2 converge and
>>>>>>>> that the sum of integers diverges, and that periodic
>>>>>>>> functions do neither.
>>>>>>>>
>>>>>>>> I.e. the criteria of convergence and existence of a limit,
>>>>>>>> and the special case "diverges as going to infinity",
>>>>>>>> are not yet closed in today's mathematics.
>>>>>>>>
>>>>>>>> It's sort of a conjecture or independent whether
>>>>>>>> they ever could be, closed, and complete, the criteria.
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>> I have spent 20 years on The conventional halting problem proofs,
>>>>>>> Gödel 1931 Incompleteness and the Liar Paradox. I have only spent
>>>>>>> about seven years on Tarski Undefinability.
>>>>>>>
>>>>>>> The Halting problem is the only one of these that has a formal
>>>>>>> system
>>>>>>> that cannot have hidden gaps its its reasoning. If a machine can
>>>>>>> do it
>>>>>>> then it can be done.
>>>>>>>
>>>>>>
>>>>>> "Algorithmics" is a study of algorithms. The most usual
>>>>>> consideration is "asymptotics", "asymptotics of complexity",
>>>>>> with the idea that computing does work, that there are
>>>>>> closures and completions, and there are approximations and
>>>>>> attenuations, as it were, in the full and the final and
>>>>>> the initial and the partial.
>>>>>>
>>>>>> The most usual resources are time, and space, that actions
>>>>>> occur in time, and according to a structure, in space.
>>>>>>
>>>>>> Of course any CS grad pretty much knows that.
>>>>>>
>>>>>>
>>>>>> The objects of mathematics, have it so, that the finite
>>>>>> and bounded resources of any collection of digital and
>>>>>> binary machines, are finite and bounded, while, the unbounded
>>>>>> models of mathematics that represent them, are unbounded,
>>>>>> it's bad that anybody ever said a Turing tape was "infinite",
>>>>>> except that it's unbounded as an object of mathematics, and
>>>>>> the model of any finite, digital, binary machine.
>>>>>>
>>>>>> One might aver "digital" has a complement in computing,
>>>>>> "analog", which represents the continuous instead of
>>>>>> the discrete, and one might aver "binary" has a complement,
>>>>>> either "real-valued" or "fuzzy" or variously about the
>>>>>> "multi-valent", in logic, the models, according to the
>>>>>> mechanisms, the models, their mechanics, machines.
>>>>>>
>>>>>> The Turing machine though is a model of unbounded machine
>>>>>> as what any thusly simply mechanical model can implement,
>>>>>> up to its bounds.
>>>>>>
>>>>>>
>>>>>> So, the objects of mathematics, then involve analytical
>>>>>> results. I can't draw a circle, though a compass may
>>>>>> portray a diagram arbitrarily close, and no different.
>>>>>> It's the analytical results, that make so for why
>>>>>> mathematics is: "sensible, fungible, and tractable",
>>>>>> and for "the ubiquitous success of mathematics" in
>>>>>> all such matters of physics, which is a continuum mechanics,
>>>>>> and as well discrete mechanics.
>>>>>>
>>>>>> Of course, results exist in mathematical objects,
>>>>>> after the unbounded the unbounded and complete,
>>>>>> which is why closures and completions are so
>>>>>> relevant to all such matters, vis-a-vis approximations
>>>>>> and attenuations, which in bounded terms result
>>>>>> indistinguishably non-different results,
>>>>>> up to precision, say, or tolerances.
>>>>>>
>>>>>> So, "machine" might be discrete, binary, and digital,
>>>>>> or it might be continuous, multivalent, and analog.
>>>>>>
>>>>>> The Turing machine in a concrete form, is bounded.
>>>>>>
>>>>>> Similar models, here mostly for the stack models,
>>>>>> stack machines, among finite-state-machines then
>>>>>> for example concepts like "unbounded nondeterminism",
>>>>>> here is pretty much about determinism, about the
>>>>>> defined behavior of a machine according to a
>>>>>> mathematical model that given defined behavior
>>>>>> of the mechanisms results a mechanical model of
>>>>>> a mathematical model, such a theory as this.
>>>>>>
>>>>>> So, the theory of computing, with mechanical or
>>>>>> computational models, models of computing, is
>>>>>> pretty much exactly like the theory of physics,
>>>>>> with the attachment of physical models to the
>>>>>> mathematical models, and rather, "interpreted as",
>>>>>> the models, the models, with respect to each other.
>>>>>>
>>>>>> I.e. the extensionality that so follows "they're
>>>>>> equal, or same, or not different, they represent
>>>>>> each other, they're equi-interpretable, they model
>>>>>> each other", the models, of logical and mathematical
>>>>>> and computational and mechanical and physical models,
>>>>>> help represent that over all the entire thing there
>>>>>> is a usual theory for the entire thing, it's a theory
>>>>>> where model theory models extensionality, and in identity
>>>>>> intensionality, about equality, tautology, identity,
>>>>>> and sameness/difference, and nearness/farness, and all
>>>>>> these usual aspects of the mathematical models'
>>>>>> arithmetic, algebra, and geometry. (And function
>>>>>> theory and topology, with respect to categories and
>>>>>> types, sets and parts, relations and predicates,
>>>>>> then all the model-building among that which is
>>>>>> all the propositional or the stipulated or axiomatic.)
>>>>>>
>>>>>> The idea is that this sort of platonistic universe of
>>>>>> mathematical objects has always existed, and it's just
>>>>>> discovered, then that axiomatics for example, just
>>>>>> sort of results a model theory of it, with regards
>>>>>> to models, the modular, modulation, modality, and the mode.
>>>>>>
>>>>>>
>>>>>> So, a machine, with respect to computation, establishing
>>>>>> the validity of interpretations of models, is subject to
>>>>>> filling out all the above, vis-a-vis what it can do,
>>>>>> and it can-do.
>>>>>>
>>>>>>
>>>>>> Then, "slowly converging" and "slowly diverging"
>>>>>> are examples that get get into that there are more
>>>>>> law(s) of large numbers, than the usual classical
>>>>>> law of large numbers.
>>>>>>
>>>>>> Some people variously do or don't have a mathematical
>>>>>> model of larger numbers, or the infinite, at all.
>>>>>> It's a totally ancient and dogmatic tradition that
>>>>>> no, we finite peoples, persons, or agents of limited
>>>>>> and finite means, no we do not have discrete, digital,
>>>>>> binary mechanical models of the infinite.
>>>>>>
>>>>>> Yet, it's also about the first thing that deduction
>>>>>> arrives at just from the plain contemplation of the
>>>>>> beyond the unbounded, like the theories of Democritus
>>>>>> and Atomism or the theories of Zeno and Continuity,
>>>>>> that make it so we at least have something to arrive
>>>>>> at for models of the unbounded, as "methods of exhaustion",
>>>>>> the approximation and attenuative what results
>>>>>> the asymptotics of algorithmics.
>>>>>>
>>>>>> So, we do have mental models of the continuous and infinite.
>>>>>> And, where physics is a continuum mechanics,
>>>>>> there are physical models of it as well, then
>>>>>> with regards to the philosophy of science and
>>>>>> the scientific method and the Big Science and
>>>>>> from the Primary Science, lots to establish
>>>>>> that the continuous and analog has mathematical
>>>>>> results, as whethe, "sensible, fungible, and tractable",
>>>>>> to machines at all.
>>>>>>
>>>>>> Numerical methods then, and, approximation and
>>>>>> attenuation, result from analysis, why they exist,
>>>>>> here for the criteria of convergence and existence
>>>>>> of limits, in the closed, in the complete.
>>>>>>
>>>>>>
>>>>>> So, a metonymy, unless it's "the Strong Metonymy",
>>>>>> which is utter intensionality of "the ground model",
>>>>>> is yet a metaphor and a metaphor yet a weaker metonymy
>>>>>> joint among weaker metonymys, that, ontology, has
>>>>>> that there are or may be a completion, of ontology,
>>>>>> as a "strong ontology" and "strong metonymy",
>>>>>> but otherwise it's just a bag of facts.
>>>>>>
>>>>>>
>>>>>>
>>>>>> Here then there's certainly a perceived analytical
>>>>>> development Cantor space, and, Square Cantor space,
>>>>>> and, Signal Cantor space as it were, about that among
>>>>>> the law(s) of large numbers, there are definitions of
>>>>>> continuity, at least including field-reals, line-reals,
>>>>>> and signal-reals, three sorts continuous domains.
>>>>>>
>>>>>>
>>>>>> Mathematics owes physics this to better begin to
>>>>>> explain, to disclose, to find truths of, continuum mechanics.
>>>>>>
>>>>>>
>>>>>> Then that analog, fuzzy, multi-value, "quantum" computing
>>>>>> models (parts) of that, that discrete, binary, digital
>>>>>> computing does not, is a thing.
>>>>>>
>>>>>>
>>>>>>
>>>>>> The convergence of terms in series is an example
>>>>>> of a model of things, that, sometimes has clear
>>>>>> and direct and perfectly accurate representations,
>>>>>> models, in stack machine, and sometimes doesn't.
>>>>>>
>>>>>>
>>>>>> Of course, for any bounded input, there is a sufficiently
>>>>>> larger bounded analyzer, that does solve it. So, why not
>>>>>> just put those all together? It would be infinite,
>>>>>> yet most things are not. (Or, you know, are.)
>>>>>>
>>>>>>
>>>>>> It's called "foundations" the study of all these things
>>>>>> as a fundamental coherency, a thing together, while of
>>>>>> its parts.
>>>>>>
>>>>>> So, students should well have the concepts of "abacus"
>>>>>> and "slide rule" and other "tabulators and computers"
>>>>>> of the usual manual mechanical variety down, according
>>>>>> to the principles of the mathematical models behind them,
>>>>>> then introducing the law(s) of large numbers, then
>>>>>> introducing the idea of a standard model of an infinite,
>>>>>> about rates, related rates, asymptotics, and bounds.
>>>>>>
>>>>>>
>>>>>> "Foundations" then the idea is that it's been around
>>>>>> forever, it's our dogma and from our canon and it develops
>>>>>> over time, and the current edition is called "modern".
>>>>>
>>>>> Did I make any mistakes?
>>>>>
>>>>> Tarski anchors his proof in the Liar Paradox
>>>>> https://liarparadox.org/Tarski_247_248.pdf
>>>>>
>>>>> How Tarski encodes the Liar Paradox
>>>>> x ∉ True if and only if p
>>>>> where the symbol 'p' represents the whole sentence x
>>>>>
>>>>> This is transformed into line 01 of his proof
>>>>> by replacing True with Provable
>>>>>
>>>>> *Here is the Tarski Undefinability Theorem proof*
>>>>> (1) x ∉ Provable if and only if p   // assumption
>>>>> (2) x ∈ True if and only if p       // assumption
>>>>> (3) x ∉ Provable if and only if x ∈ True.
>>>>> (4) either x ∉ True or x̄ ∉ True;    // axiom: ~True(x) ∨ ~True(~x)
>>>>> (5) if x ∈ Provable, then x ∈ True; // axiom: Provable(x) → True(x)
>>>>> (6) if x̄ ∈ Provable, then x̄ ∈ True;  // axiom: Provable(~x) → True(~x)
>>>>> (7) x ∈ True
>>>>> (8) x ∉ Provable
>>>>> (9) x̄ ∉ Provable
>>>>>
>>>>> Tarski's Full proof
>>>>> https://liarparadox.org/Tarski_275_276.pdf
>>>>>
>>>>>
>>>>>
>>>>
>>>>
>>>> Well I have some collected works of Tarski so
>>>> I'll be looking to them, the excerpt there you
>>>> reference basically talks about the model under
>>>> consideration, as a fragment, of "the meta-theory",
>>>> i.e., "Tarski's meta-theory" is "rather underdefined",
>>>> that what he does is that he says that in the meta-theory,
>>>> that there's something about the model under consideration,
>>>> that isn't true in the model but is true, and he invokes
>>>> Goedel to basically says that Goedel's incompleteness
>>>> applies to this model, which of course must thusly by
>>>> "strong enough to support arithmetic", where otherwise
>>>> of course, Goedel's incompleteness does _not_ apply
>>>> to theories that may be entirely closed categories.
>>>>
>>>> It seems the issue is that you think that you have
>>>> entirely closed categories, in your meta-theory,
>>>> then are applying this meta-theory to another model
>>>> under consideration, which isn't, so it seems like
>>>> you have implicits on your model, which are underdefined,
>>>> rather as the "properly logical" or "non-logical" are
>>>> with respect to being modeled by the logical, that
>>>> you have the some sort of "un-logical" that are the
>>>> result of that the axioms are un-logical, with regards
>>>> to that your "meta-theory" is logical and your model
>>>> under consideration, its _logical_ objects, are actually
>>>> "un-logical" objects with respect to your, "meta-theory".
>>>>
>>>>
>>>> That's a problem overall with "meta-theory", here the
>>>> goal is to arrive at a theory that is its own meta-theory,
>>>> that's also all one language, with regards to otherwise
>>>
>>> Yes that is exactly correct.
>>>
>>> *True(L,x) returns TRUE when x is True otherwise returns FALSE*
>>
>> So what does True(L, S) return when S is defined as S := ~True(L,S)
>>
>
> True(L, S) returns Prolog's negation as failure false
> that only means not true.