On 1/23/2023 4:54 PM, Richard Damon wrote:
> On 1/23/23 5:23 PM, olcott wrote:
>> On 1/23/2023 10:51 AM, Richard Damon wrote:
>>> On 1/23/23 10:18 AM, olcott wrote:
>>>> On 1/20/2023 4:54 PM, Richard Damon wrote:
>>>>> On 1/20/23 5:16 PM, olcott wrote:
>>>>>> On 1/20/2023 4:09 PM, Richard Damon wrote:
>>>>>>> On 1/20/23 5:02 PM, olcott wrote:
>>>>>>>> On 1/20/2023 2:46 PM, Richard Damon wrote:
>>>>>>>>> On 1/20/23 2:31 PM, olcott wrote:
>>>>>>>>>> On 1/19/2023 8:34 PM, Richard Damon wrote:
>>>>>>>>>>> On 1/19/23 2:12 PM, olcott wrote:
>>>>>>>>>>>> On 1/17/2023 5:44 PM, Richard Damon wrote:
>>>>>>>>>>>>> On 1/17/23 11:39 AM, olcott wrote:
>>>>>>>>>>>>>> On 1/16/2023 7:51 PM, Richard Damon wrote:
>>>>>>>>>>>>>>> No, because I am showing that G is TRUE, not PROVABLE.
>>>>>>>>>>>>>>> Truth can use infinte sets oc connections, proofs can't.
>>>>>>>>>>>>>>> Only YOU have perposed that we think about infinite proofs.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Formal systems cannot ever use infinite connections from
>>>>>>>>>>>>>> their
>>>>>>>>>>>>>> expressions of language to their truth maker axioms thus
>>>>>>>>>>>>>> eliminating
>>>>>>>>>>>>>> these from consideration as any measure of true "in the
>>>>>>>>>>>>>> system".
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> Source? or is this just another of your made up "Facts"
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> You can't even remember that you said this?
>>>>>>>>>>>
>>>>>>>>>>> No, I said they can't have infinite PROOFS, not infinite
>>>>>>>>>>> connections to Truth.
>>>>>>>>>>>>
>>>>>>>>>>>>> WHERE in the definition of a "Formal System" does it say
>>>>>>>>>>>>> that the connecti0on must be finite.
>>>>>>>>>>>>
>>>>>>>>>>>> You said that formal system cannot have infinite proofs.
>>>>>>>>>>>> Did you change your mind?
>>>>>>>>>>>
>>>>>>>>>>> Right ***PROOF*** not ***TRUTH***
>>>>>>>>>>>
>>>>>>>>>>> Truth can be based on an infinite chain of connections,
>>>>>>>>>>> proofs can not.
>>>>>>>>>>
>>>>>>>>>> Truth *in a formal system* cannot be based on infinite
>>>>>>>>>> connections because formal systems are not allowed to have
>>>>>>>>>> infinite connections.
>>>>>>>>>
>>>>>>>>> Says Who ***FOR TRUTH***
>>>>>>>>>
>>>>>>>>> You reference does not provide that data, so I guess you are
>>>>>>>>> just making it up, and thus showing you to be a LIAR.
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Haskell Curry establishes that truth in a theory (AKA formal
>>>>>>>>>> system) is anchored in the elementary theorems (AKA axioms) of
>>>>>>>>>> this formal system.
>>>>>>>>>
>>>>>>>>> Right, ANCHORED TO, not limited to. Statments other than the
>>>>>>>>> elementary theorems are True, and they are true if they have a
>>>>>>>>> connection (not limited to finite) to these Truths.
>>>>>>>>>
>>>>>>>>> Where does he say True statements must have a FINITE connection
>>>>>>>>> to the elementary theorems.
>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> A theory (over (f) is defined as a conceptual class of these
>>>>>>>>>> elementary
>>>>>>>>>> statements. Let::t be such a theory. Then the elementary
>>>>>>>>>> statements
>>>>>>>>>> which belong to ::t we shall call the elementary theorems
>>>>>>>>>> of::t; we also
>>>>>>>>>> say that these elementary statements are true for::t. Thus,
>>>>>>>>>> given ::t,
>>>>>>>>>> an elementary theorem is an elementary statement which is
>>>>>>>>>> true. A theory
>>>>>>>>>> is thus a way of picking out from the statements of (f a certain
>>>>>>>>>> subclass of true statements.
>>>>>>>>>> https://www.liarparadox.org/Haskell_Curry_45.pdf
>>>>>>>>>>
>>>>>>>>>> Perhaps you believe that you are enormously much brighter than
>>>>>>>>>> Haskell Curry ?
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> You don't understand what he is saying,
>>>>>>>>>
>>>>>>>>> He is saying these statements are True in F, as a given.
>>>>>>>>
>>>>>>>> Wrongo !!!
>>>>>>>>
>>>>>>>>     The terminology which has just been used implies that the
>>>>>>>>     elementary statements are not such that their truth and
>>>>>>>>     falsity are known to us without reference to::t.
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>> Right, they aren't just true in the Statement class, but are only
>>>>>>> considerdd true because we are in the Theory F.
>>>>>>>
>>>>>>
>>>>>> F is not the theory T is the theory.
>>>>>>
>>>>>>
>>>>>
>>>>>
>>>>> Red Herring.
>>>>>
>>>>> F is the Theory in Godels descussion.
>>>>>
>>>>
>>>> https://en.wikipedia.org/wiki/Metamathematics
>>>> LP := ~True(LP) is untrue yet that does not make it true.
>>>>
>>>> When we examine this at the meta level we escape the self-contradiction
>>>> and can say that it is true that LP is untrue.
>>>
>>>
>>> Excpet that untrue is not ~True() in classical logic, which makes
>>> statements either True or False, or makes them Not a Truth Bearer,
>>> which makes them not in the domain of the True predicate.
>>>
>>> You need to move to tri-value logic to do this, at which point you
>>> loose the relationship that ~True(x) -> False(x)
>>>
>>
>> True / false and not a truth bearer.
>
> That is your TRI-value logic.
>

It is true by logical necessity.
Every expression of language must necessarily be
true, false, neither true nor false.

>>
>>> Note, most of mathematics is based on the two-value logic system.
>>>
>>>
>>
>> Thus forcing it to classify "not a truth bearer" incorrectly.
>> If all you have is a hammer the unscrewing a screw becomes quite
>> destructive.
>
> Nope, a "statement" can be well formed, and thus MUST be a "Truth
> Bearer" or it isn't and is NOT a "Truth Bearer"
>
> By ignoring that mathematically defined statement ARE "Truth Bearers",
> you logic system is just broken.
>
>>
>>>>
>>>> https://plato.stanford.edu/entries/tarski-truth/#195DefOff
>>>>
>>>> It looks like model theory is required to determine the truth of
>>>> some mathematical expressions, this had it origins in Tarski's
>>>> definition of truth.
>>>>
>>>> ∃n ∈ ℕ (N > 3)       // does not seem to need model theory
>>>> ∃G ∈ F (G ↔ (F ⊬ G)) // does not seem to need model theory
>>>>
>>>
>>> ∃ is a symbol out of model theory, so hard to not need model theory.
>>>
>>> Quoting from your reference:
>>>
>>> Model theory by contrast works with three levels of symbol. There are
>>> the logical constants ( = , ¬ , & for example), the variables (as
>>> before), and between these a middle group of symbols which have no
>>> fixed meaning but get a meaning through being applied to a particular
>>> structure. The symbols of this middle group include the nonlogical
>>> constants of the language, such as relation symbols, function symbols
>>> and constant individual symbols. They also include the quantifier
>>> symbols  ∀ and ∃, since we need to refer to the structure to see what
>>> set they range over.
>>
>> I just showed how to explicitly specify what they range over: ∃n ∈ ℕ
>
> Which means you are using "Model Theory"
>
> Maybe you don't understand those words.

Model theory is used to define things that are not otherwise defined.
When they are otherwise defined there is no need for model theory.

>>
>>>> G is true in F iff it cannot be shown that G is true in F
>>>>
>>>
>>> Nope, you don't understand what G is. The Definition of G in F does
>>> NOT refer in any way determinable in F to the statement G.
>>>
>>
>> ∃n ∈ ℕ (n > 3) // Is this true or false?
>> How do you know?
>
> Simple, 4 exists (S(S(S(S(0)))), 4 > 3, 4 ∈ ℕ, thus the statement is
> True. Like many (but not all) True statements, it can be proven.
>
>>
>> Generically how does ascertain that that any logic expression is true or
>> false?
>
> Note, "Ascertain" means you are talking about KNOWLEDGE, not Truth.
>
> Truth doesn't need to be ascertained to be true, it just is.
>
> It needs to be ascertained to be KNOWN.
>
> It is a TRUE statement that either all even numbers greater than 2 are
> the sum of 2 primes or there exists at least one that is not. We don't
> know which one of them is true right now, but we do know that one of
> them is.
>
> This seems to be one of your core problems, confusing what can be known
> to be true with what IS true.
>
>>
>> Most generically an analytical expression of formal or natural
>> language is only true if it has a semantic connection to its truth
>> maker axioms.
>
> Right, but that connection might not be known, or might even be infinite.
>
> It is only KNOWLEDGE or PROOF that requires a finite connection.
>>
>> The "truth maker axioms" of natural language are the definition of the
>> meaning of its words.
>
>
> No, the accepted Truth Maker Axioms of the Theory (not what their words
> mean in Natural Language) determine what is true.

of natural language such as English
of natural language such as English
of natural language such as English
of natural language such as English

>
> Your reliance on "Natural Language" is what has actually been proven to
> lead to problems.
>

The entire body of all analytical knowledge can only be expressed using
language. Hardly any of this is currently expressed using formal
language. All knowledge is necessarily true by definition.

>>
>> The truth maker axioms for the above expression is the definition of
>> the ordered set of natural numbers:
>>
>> https://www.britannica.com/science/Peano-axioms
>>
>
> You understand that Godel showed that under the Peano-axioms, he proved
> that their exists truths that can not be proven.

We can make the Gödel number of "I just ate some chicken" using the
adjacent ASCII values. This too cannot be proven in the Peano-axioms.

> It becomes a
> consequence of the induction axiom that allows him to be able to define
> the primative recursive relationship that shows that you can not prove
> within the theory that no nmber exists that matches that theory, and
> also create an extention to that theory (that is used to create that
> relationship) that allows us to actually prove that statment must be
> true, and also that no proof of this can exist in the base theory.
>

Its a mere gimmick.
He acknowledged that the Liar Paradox forms an equivalent proof.

> The induction property that proves it only comes in the extension (the
> meta-theory) and is not in the base theory,

"This sentence is not true" is self-evidently untrue yet that does not
make the sentence true within the scope of self-contradiction.

> so the base theory can't
> make the proof, but can evaluate for every term, thus making the
> INFINITE chain that makes it true in the Theory.
>

It is not an infinite chain, it is simply that the sentence is true
outside of the scope of self-contradiction and impossible to evaluate
within the scope of self-contradiction.

> Peano ARITHMATIC changed that induction axiom to a first order logic
> definition, weaking what the theory can do, but allows it to appear to
> be complete, but NOT express ALL the properties of the Natural Numbers.
>

Natural numbers themselves never had the property of provability.

*The five Peano axioms are*
(1) Zero is a natural number.
(2) Every natural number has a successor in the natural numbers.
(3) Zero is not the successor of any natural number.
(4) If the successor of two natural numbers is the same, then the two
original numbers are the same.
(5) If a set contains zero and the successor of every number is in the
set, then the set contains the natural numbers.

https://www.britannica.com/science/Peano-axioms

> I beleive that it becomes the (or at least one of the) largest logic
> system that retains "Completeness" while sitll being "Consistent". (But
> it can't prove itself to be consistent)
>
> This is of course, over you head, so you wil  either deny it or just
> ignore the refuation and go off on some other tack.

--
Copyright 2023 Olcott

"Talent hits a target no one else can hit;
Genius hits a target no one else can see."
Arthur Schopenhauer