Theorem 1.
Every valid logical expression is provable. Equivalently, every logical
expression is either satisfiable or refutable.
https://plato.stanford.edu/entries/goedel/#ComThe

The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))

Should actually be written as:
Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
thus abolishing Incompleteness.

--
Copyright 2022 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 12/10/22 5:59 PM, olcott wrote:
> Theorem 1.
> Every valid logical expression is provable. Equivalently, every logical
> expression is either satisfiable or refutable.
> https://plato.stanford.edu/entries/goedel/#ComThe
>
> The conventional definition of incompleteness:
> Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
>
> Should actually be written as:
> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
> thus abolishing Incompleteness.
>
>

Did you notice what domain that Theorem was stated for?

"First Order" deals with just a small limited subset of Mathematics.

Yes, it is know that First Orders system can be complete, but can not
express all the known properties of the Natural Numbers.

Comutation Theory is NOT a "First Order Predicate Calculus" (it uses
Higher Order logical predicated), so that Theorem doesn't apply.

On 12/10/2022 8:18 PM, Richard Damon wrote:
> On 12/10/22 5:59 PM, olcott wrote:
>> Theorem 1.
>> Every valid logical expression is provable. Equivalently, every
>> logical expression is either satisfiable or refutable.
>> https://plato.stanford.edu/entries/goedel/#ComThe
>>
>> The conventional definition of incompleteness:
>> Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
>>
>> Should actually be written as:
>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
>> thus abolishing Incompleteness.
>>
>>
>
> Did you notice what domain that Theorem was stated for?
>
> "First Order" deals with just a small limited subset of Mathematics.
>
> Yes, it is know that First Orders system can be complete, but can not
> express all the known properties of the Natural Numbers.
>
> Comutation Theory is NOT a "First Order Predicate Calculus" (it uses
> Higher Order logical predicated), so that Theorem doesn't apply.

None-the-less its still applies to the philosophical foundation of every
formal system. If an expression of formal or natural language is neither
provable nor refutable then it is not a logical expression of any kind
at all.

Everyone that has been debating me on this forum does so entirely on the
basis of learn-by-rote dogma.

No one has considered that this math dogma might be inconsistent thus
incoherent when examined through its foundation basis.

--
Copyright 2022 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 12/10/22 6:33 PM, olcott wrote:
> On 12/10/2022 8:18 PM, Richard Damon wrote:
>> On 12/10/22 5:59 PM, olcott wrote:
>>> Theorem 1.
>>> Every valid logical expression is provable. Equivalently, every
>>> logical expression is either satisfiable or refutable.
>>> https://plato.stanford.edu/entries/goedel/#ComThe
>>>
>>> The conventional definition of incompleteness:
>>> Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
>>>
>>> Should actually be written as:
>>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
>>> thus abolishing Incompleteness.
>>>
>>>
>>
>> Did you notice what domain that Theorem was stated for?
>>
>> "First Order" deals with just a small limited subset of Mathematics.
>>
>> Yes, it is know that First Orders system can be complete, but can not
>> express all the known properties of the Natural Numbers.
>>
>> Comutation Theory is NOT a "First Order Predicate Calculus" (it uses
>> Higher Order logical predicated), so that Theorem doesn't apply.
>
> None-the-less its still applies to the philosophical foundation of every
> formal system. If an expression of formal or natural language is neither
> provable nor refutable then it is not a logical expression of any kind
> at all.
>

Nope. Shows you don't understand what you are talking about,

Note, it isn't that True is DEFINED as provable, but that for many first
order systems, it can be proven that all true statements can be proven.

Yes, it is possible to include in a system an Axiom/Definition that says
that all True Statements are Provable, and such a system needs to limit
its logical capability, like it can only use first order predicates, or
it will become inconsistent, and such inconsistancy might not be
detectable within the system.

Such a system can NOT handle the fullness of Mathematics, or systems
like Computation Theory

> Everyone that has been debating me on this forum does so entirely on the
> basis of learn-by-rote dogma.

Nope, you have been arguing from never-learned-due-to-stupdity dogma.

Your IGNORANCE does make your statements true.

>
> No one has considered that this math dogma might be inconsistent thus
> incoherent when examined through its foundation basis.
>

No, you are just showing that you don't understanf what you are talking
about.

On Sunday, 11 December 2022 at 03:59:47 UTC+2, olcott wrote:
> Should actually be written as:
> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
Is this a 1st order; or high-order sentence?

On Sunday, 11 December 2022 at 04:33:32 UTC+2, olcott wrote:
> No one has considered that this math dogma might be inconsistent thus
> incoherent when examined through its foundation basis.

Nobody cares if Mathematics is inconsistent because it's useful.

On 12/11/2022 2:09 AM, Skep Dick wrote:
> On Sunday, 11 December 2022 at 04:33:32 UTC+2, olcott wrote:
>> No one has considered that this math dogma might be inconsistent thus
>> incoherent when examined through its foundation basis.
>
> Nobody cares if Mathematics is inconsistent because it's useful.
>
If mathematics is not incomplete or inconsistent then the Tarski
Undefinability isomorphism of the incompleteness theorem can be rejected
thus providing the anchor to truth conditional semantics. We can't get
very far with AI research without this anchor.

--
Copyright 2022 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 12/11/2022 2:00 AM, Skep Dick wrote:
> On Sunday, 11 December 2022 at 03:59:47 UTC+2, olcott wrote:
>> Should actually be written as:
>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
> Is this a 1st order; or high-order sentence?

Every order from first to N.

--
Copyright 2022 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 12/10/2022 10:19 PM, Richard Damon wrote:
> On 12/10/22 6:33 PM, olcott wrote:
>> On 12/10/2022 8:18 PM, Richard Damon wrote:
>>> On 12/10/22 5:59 PM, olcott wrote:
>>>> Theorem 1.
>>>> Every valid logical expression is provable. Equivalently, every
>>>> logical expression is either satisfiable or refutable.
>>>> https://plato.stanford.edu/entries/goedel/#ComThe
>>>>
>>>> The conventional definition of incompleteness:
>>>> Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
>>>>
>>>> Should actually be written as:
>>>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
>>>> thus abolishing Incompleteness.
>>>>
>>>>
>>>
>>> Did you notice what domain that Theorem was stated for?
>>>
>>> "First Order" deals with just a small limited subset of Mathematics.
>>>
>>> Yes, it is know that First Orders system can be complete, but can not
>>> express all the known properties of the Natural Numbers.
>>>
>>> Comutation Theory is NOT a "First Order Predicate Calculus" (it uses
>>> Higher Order logical predicated), so that Theorem doesn't apply.
>>
>> None-the-less its still applies to the philosophical foundation of every
>> formal system. If an expression of formal or natural language is neither
>> provable nor refutable then it is not a logical expression of any kind
>> at all.
>>
>
> Nope. Shows you don't understand what you are talking about,
>
> Note, it isn't that True is DEFINED as provable, but that for many first
> order systems, it can be proven that all true statements can be proven.
>
> Yes, it is possible to include in a system an Axiom/Definition that says
> that all True Statements are Provable, and such a system needs to limit
> its logical capability, like it can only use first order predicates, or
> it will become inconsistent, and such inconsistancy might not be
> detectable within the system.
>
> Such a system can NOT handle the fullness of Mathematics, or systems
> like Computation Theory
>

It handles the subset of computations that are decision problems through
the Curry Howard Isomorphism.

>> Everyone that has been debating me on this forum does so entirely on the
>> basis of learn-by-rote dogma.
>
> Nope, you have been arguing from never-learned-due-to-stupdity dogma.
>
> Your IGNORANCE does make your statements true.
>
>>
>> No one has considered that this math dogma might be inconsistent thus
>> incoherent when examined through its foundation basis.
>>
>
> No, you are just showing that you don't understanf what you are talking
> about.

The conventional definition of incompleteness:
Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))

is wrong the correct way to say this is
Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))

--
Copyright 2022 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On Sunday, 11 December 2022 at 15:48:27 UTC+2, olcott wrote:
> On 12/11/2022 2:00 AM, Skep Dick wrote:
> > On Sunday, 11 December 2022 at 03:59:47 UTC+2, olcott wrote:
> >> Should actually be written as:
> >> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
> > Is this a 1st order; or high-order sentence?
> Every order from first to N.
ℕ is not a number.

No idea what you mean by 1 to ℕ

On Sunday, 11 December 2022 at 15:53:29 UTC+2, olcott wrote:
> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
So which one is it?

The disjunction requires a valid proof.

On 12/11/22 6:42 AM, Skep Dick wrote:
> On Sunday, 11 December 2022 at 15:48:27 UTC+2, olcott wrote:
>> On 12/11/2022 2:00 AM, Skep Dick wrote:
>>> On Sunday, 11 December 2022 at 03:59:47 UTC+2, olcott wrote:
>>>> Should actually be written as:
>>>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
>>> Is this a 1st order; or high-order sentence?
>> Every order from first to N.
> ℕ is not a number.
>
> No idea what you mean by 1 to ℕ

No, you are a DICK,

He said 1 to N, not to ℕ

N is clasically a counting number.

while ℕ is classically the name for the Natural Numbers.

You are just showing you have a worse reguard for Truth then Olcott
does. He seems to at least beleive what he babbles about, it seems that
you are just intentionally misinterpreting people by choice.

On Sunday, 11 December 2022 at 15:53:29 UTC+2, olcott wrote:
> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
The way you prove a disjunction is straight out of the BHK interpretation....

https://en.wikipedia.org/wiki/Brouwer%E2%80%93Heyting%E2%80%93Kolmogorov_interpretation

A proof of P ∨ Q is either <0,a> where a is a proof of P or <1,b> where b is a proof of Q.

On Sunday, 11 December 2022 at 16:47:54 UTC+2, richar...@gmail.com wrote:
> On 12/11/22 6:42 AM, Skep Dick wrote:
> > On Sunday, 11 December 2022 at 15:48:27 UTC+2, olcott wrote:
> >> On 12/11/2022 2:00 AM, Skep Dick wrote:
> >>> On Sunday, 11 December 2022 at 03:59:47 UTC+2, olcott wrote:
> >>>> Should actually be written as:
> >>>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
> >>> Is this a 1st order; or high-order sentence?
> >> Every order from first to N.
> > ℕ is not a number.
> >
> > No idea what you mean by 1 to ℕ
> No, you are a DICK,
>
> He said 1 to N, not to ℕ
>
> N is clasically a counting number.
Idiot. He is talking about ordinals.

What is the domain of the ordinals 1 to N ?

On 12/11/22 5:53 AM, olcott wrote:
> On 12/10/2022 10:19 PM, Richard Damon wrote:
>> On 12/10/22 6:33 PM, olcott wrote:
>>> On 12/10/2022 8:18 PM, Richard Damon wrote:
>>>> On 12/10/22 5:59 PM, olcott wrote:
>>>>> Theorem 1.
>>>>> Every valid logical expression is provable. Equivalently, every
>>>>> logical expression is either satisfiable or refutable.
>>>>> https://plato.stanford.edu/entries/goedel/#ComThe
>>>>>
>>>>> The conventional definition of incompleteness:
>>>>> Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
>>>>>
>>>>> Should actually be written as:
>>>>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
>>>>> thus abolishing Incompleteness.
>>>>>
>>>>>
>>>>
>>>> Did you notice what domain that Theorem was stated for?
>>>>
>>>> "First Order" deals with just a small limited subset of Mathematics.
>>>>
>>>> Yes, it is know that First Orders system can be complete, but can
>>>> not express all the known properties of the Natural Numbers.
>>>>
>>>> Comutation Theory is NOT a "First Order Predicate Calculus" (it uses
>>>> Higher Order logical predicated), so that Theorem doesn't apply.
>>>
>>> None-the-less its still applies to the philosophical foundation of every
>>> formal system. If an expression of formal or natural language is neither
>>> provable nor refutable then it is not a logical expression of any kind
>>> at all.
>>>
>>
>> Nope. Shows you don't understand what you are talking about,
>>
>> Note, it isn't that True is DEFINED as provable, but that for many
>> first order systems, it can be proven that all true statements can be
>> proven.
>>
>> Yes, it is possible to include in a system an Axiom/Definition that
>> says that all True Statements are Provable, and such a system needs to
>> limit its logical capability, like it can only use first order
>> predicates, or it will become inconsistent, and such inconsistancy
>> might not be detectable within the system.
>>
>> Such a system can NOT handle the fullness of Mathematics, or systems
>> like Computation Theory
>>
>
> It handles the subset of computations that are decision problems through
> the Curry Howard Isomorphism.

No, Computation Theory uses logic operations not allowed withinvsystems
that meet the requirements of the Completeness Theorem.

Curry Howard also only applies in systems that HAVE programs, which
aren't the sort of systems that the Completeness Theorem holds for.

You are just showing you don't understand the concept of
pre-requirements of a Theory.

>
>>> Everyone that has been debating me on this forum does so entirely on the
>>> basis of learn-by-rote dogma.
>>
>> Nope, you have been arguing from never-learned-due-to-stupdity dogma.
>>
>> Your IGNORANCE does make your statements true.
>>
>>>
>>> No one has considered that this math dogma might be inconsistent thus
>>> incoherent when examined through its foundation basis.
>>>
>>
>> No, you are just showing that you don't understanf what you are
>> talking about.
>
> The conventional definition of incompleteness:
> Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
>
> is wrong the correct way to say this is
> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
>

No, your "Valid" is just a restatement of Complete.

I don't know of a use of the Valid as applying to SYSTEMS, just
arguments/proofs. The closes to Validity is Consistent, which means that
no statement can be proven both True and False.

There is nothing "Wrong" with an incomplete system that makes is not valid.

That it does't meet your INCORRECT ideas, isn't grounds to claim they are.

On 12/11/2022 8:42 AM, Skep Dick wrote:
> On Sunday, 11 December 2022 at 15:48:27 UTC+2, olcott wrote:
>> On 12/11/2022 2:00 AM, Skep Dick wrote:
>>> On Sunday, 11 December 2022 at 03:59:47 UTC+2, olcott wrote:
>>>> Should actually be written as:
>>>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
>>> Is this a 1st order; or high-order sentence?
>> Every order from first to N.
> ℕ is not a number.
>
> No idea what you mean by 1 to ℕ

I did not use the math ℕ, I used N same as n.

--
Copyright 2022 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 12/11/2022 8:44 AM, Skep Dick wrote:
> On Sunday, 11 December 2022 at 15:53:29 UTC+2, olcott wrote:
>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
> So which one is it?
>
> The disjunction requires a valid proof.
>

Unless φ is provable or refutable φ is not a logic expression.

--
Copyright 2022 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 12/11/22 6:50 AM, Skep Dick wrote:
> On Sunday, 11 December 2022 at 16:47:54 UTC+2, richar...@gmail.com wrote:
>> On 12/11/22 6:42 AM, Skep Dick wrote:
>>> On Sunday, 11 December 2022 at 15:48:27 UTC+2, olcott wrote:
>>>> On 12/11/2022 2:00 AM, Skep Dick wrote:
>>>>> On Sunday, 11 December 2022 at 03:59:47 UTC+2, olcott wrote:
>>>>>> Should actually be written as:
>>>>>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
>>>>> Is this a 1st order; or high-order sentence?
>>>> Every order from first to N.
>>> ℕ is not a number.
>>>
>>> No idea what you mean by 1 to ℕ
>> No, you are a DICK,
>>
>> He said 1 to N, not to ℕ
>>
>> N is clasically a counting number.
> Idiot. He is talking about ordinals.
>
> What is the domain of the ordinals 1 to N ?
>

Right, and ordinals are numbers (Natural Numbers in particular), so you
claim that it is not a number is just a lie.

Yes, the values of 1 to N fall as values in the Natural Numbers, which
are classically named as ℕ, but that isn't the N he was referring to.

You are just proving your incompetence at really doing logic.

On 12/11/22 7:11 AM, olcott wrote:
> On 12/11/2022 8:44 AM, Skep Dick wrote:
>> On Sunday, 11 December 2022 at 15:53:29 UTC+2, olcott wrote:
>>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
>> So which one is it?
>>
>> The disjunction requires a valid proof.
>>
>
> Unless φ is provable or refutable φ is not a logic expression.
>

Says Who? Provide a valid reference from someone knowlegable of the field.

You are using the wrong definitions of the terms.

On 12/11/2022 9:08 AM, Richard Damon wrote:
> On 12/11/22 5:53 AM, olcott wrote:
>> On 12/10/2022 10:19 PM, Richard Damon wrote:
>>> On 12/10/22 6:33 PM, olcott wrote:
>>>> On 12/10/2022 8:18 PM, Richard Damon wrote:
>>>>> On 12/10/22 5:59 PM, olcott wrote:
>>>>>> Theorem 1.
>>>>>> Every valid logical expression is provable. Equivalently, every
>>>>>> logical expression is either satisfiable or refutable.
>>>>>> https://plato.stanford.edu/entries/goedel/#ComThe
>>>>>>
>>>>>> The conventional definition of incompleteness:
>>>>>> Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
>>>>>>
>>>>>> Should actually be written as:
>>>>>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
>>>>>> thus abolishing Incompleteness.
>>>>>>
>>>>>>
>>>>>
>>>>> Did you notice what domain that Theorem was stated for?
>>>>>
>>>>> "First Order" deals with just a small limited subset of Mathematics.
>>>>>
>>>>> Yes, it is know that First Orders system can be complete, but can
>>>>> not express all the known properties of the Natural Numbers.
>>>>>
>>>>> Comutation Theory is NOT a "First Order Predicate Calculus" (it
>>>>> uses Higher Order logical predicated), so that Theorem doesn't apply.
>>>>
>>>> None-the-less its still applies to the philosophical foundation of
>>>> every
>>>> formal system. If an expression of formal or natural language is
>>>> neither
>>>> provable nor refutable then it is not a logical expression of any kind
>>>> at all.
>>>>
>>>
>>> Nope. Shows you don't understand what you are talking about,
>>>
>>> Note, it isn't that True is DEFINED as provable, but that for many
>>> first order systems, it can be proven that all true statements can be
>>> proven.
>>>
>>> Yes, it is possible to include in a system an Axiom/Definition that
>>> says that all True Statements are Provable, and such a system needs
>>> to limit its logical capability, like it can only use first order
>>> predicates, or it will become inconsistent, and such inconsistancy
>>> might not be detectable within the system.
>>>
>>> Such a system can NOT handle the fullness of Mathematics, or systems
>>> like Computation Theory
>>>
>>
>> It handles the subset of computations that are decision problems
>> through the Curry Howard Isomorphism.
>
> No, Computation Theory uses logic operations not allowed withinvsystems
> that meet the requirements of the Completeness Theorem.
>
> Curry Howard also only applies in systems that HAVE programs, which
> aren't the sort of systems that the Completeness Theorem holds for.
>
> You are just showing you don't understand the concept of
> pre-requirements of a Theory.
>

Every expression of language that cannot possibly resolve to true or
false is not a logic expression. The computation deriving the square
foot area of a house is not a logic expression.

>>
>>>> Everyone that has been debating me on this forum does so entirely on
>>>> the
>>>> basis of learn-by-rote dogma.
>>>
>>> Nope, you have been arguing from never-learned-due-to-stupdity dogma.
>>>
>>> Your IGNORANCE does make your statements true.
>>>
>>>>
>>>> No one has considered that this math dogma might be inconsistent thus
>>>> incoherent when examined through its foundation basis.
>>>>
>>>
>>> No, you are just showing that you don't understanf what you are
>>> talking about.
>>
>> The conventional definition of incompleteness:
>> Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
>>
>> is wrong the correct way to say this is
>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
>>
>
> No, your "Valid" is just a restatement of Complete.
>

You are ignoring GIGO: garbage in, garbage out.
Every self-contradictory expression of language that previously showed
that a formal system is incomplete actually shows that the expression
itself is garbage.

> I don't know of a use of the Valid as applying to SYSTEMS, just
> arguments/proofs. The closes to Validity is Consistent, which means that
> no statement can be proven both True and False.
>
> There is nothing "Wrong" with an incomplete system that makes is not valid.

When the term "incomplete" is applied to formal systems in this way it
is a God damned lie.

>
> That it does't meet your INCORRECT ideas, isn't grounds to claim they are.

--
Copyright 2022 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On Sunday, 11 December 2022 at 17:10:05 UTC+2, olcott wrote:
> > No idea what you mean by 1 to ℕ
> I did not use the math ℕ, I used N same as n.
Yes. Those are called free/unbound variables!

You are talking about the range [1..n]
What is n?

If n is unbound then [1..n] is not Valid(φ) !!!

On Sunday, 11 December 2022 at 17:12:18 UTC+2, richar...@gmail.com wrote:
> On 12/11/22 6:50 AM, Skep Dick wrote:
> > On Sunday, 11 December 2022 at 16:47:54 UTC+2, richar...@gmail.com wrote:
> >> On 12/11/22 6:42 AM, Skep Dick wrote:
> >>> On Sunday, 11 December 2022 at 15:48:27 UTC+2, olcott wrote:
> >>>> On 12/11/2022 2:00 AM, Skep Dick wrote:
> >>>>> On Sunday, 11 December 2022 at 03:59:47 UTC+2, olcott wrote:
> >>>>>> Should actually be written as:
> >>>>>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
> >>>>> Is this a 1st order; or high-order sentence?
> >>>> Every order from first to N.
> >>> ℕ is not a number.
> >>>
> >>> No idea what you mean by 1 to ℕ
> >> No, you are a DICK,
> >>
> >> He said 1 to N, not to ℕ
> >>
> >> N is clasically a counting number.
> > Idiot. He is talking about ordinals.
> >
> > What is the domain of the ordinals 1 to N ?
> >
> Right, and ordinals are numbers (Natural Numbers in particular), so you
> claim that it is not a number is just a lie.
It's not a number.

It's a free/unbound variable.

On Sunday, 11 December 2022 at 17:11:05 UTC+2, olcott wrote:
> On 12/11/2022 8:44 AM, Skep Dick wrote:
> > On Sunday, 11 December 2022 at 15:53:29 UTC+2, olcott wrote:
> >> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
> > So which one is it?
> >
> > The disjunction requires a valid proof.
> >
> Unless φ is provable or refutable φ is not a logic expression.
ERGO!!!

1 to n is NOT a logic expression. Because logic expressiong (e.g the sort of things you are referring to when you say "φ") DON'T contain free variables.

On 12/11/2022 9:15 AM, Richard Damon wrote:
> On 12/11/22 7:11 AM, olcott wrote:
>> On 12/11/2022 8:44 AM, Skep Dick wrote:
>>> On Sunday, 11 December 2022 at 15:53:29 UTC+2, olcott wrote:
>>>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
>>> So which one is it?
>>>
>>> The disjunction requires a valid proof.
>>>
>>
>> Unless φ is provable or refutable φ is not a logic expression.
>>
>
> Says Who? Provide a valid reference from someone knowlegable of the field.
>
> You are using the wrong definitions of the terms.

When everyone else uses to terms incorrectly then I seem like the black
sheep of a herd of sheep. The sheep never notice their own group-think
error. https://www.psychologytoday.com/us/basics/groupthink

Truth itself is not a Democracy. Prior to Pythagoras there was a
universal consensus that the Earth was flat. This does not make the
Earth flat it merely means that everyone was wrong.

--
Copyright 2022 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

On 12/11/22 7:17 AM, olcott wrote:
> On 12/11/2022 9:08 AM, Richard Damon wrote:
>> On 12/11/22 5:53 AM, olcott wrote:
>>> On 12/10/2022 10:19 PM, Richard Damon wrote:
>>>> On 12/10/22 6:33 PM, olcott wrote:
>>>>> On 12/10/2022 8:18 PM, Richard Damon wrote:
>>>>>> On 12/10/22 5:59 PM, olcott wrote:
>>>>>>> Theorem 1.
>>>>>>> Every valid logical expression is provable. Equivalently, every
>>>>>>> logical expression is either satisfiable or refutable.
>>>>>>> https://plato.stanford.edu/entries/goedel/#ComThe
>>>>>>>
>>>>>>> The conventional definition of incompleteness:
>>>>>>> Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
>>>>>>>
>>>>>>> Should actually be written as:
>>>>>>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
>>>>>>> thus abolishing Incompleteness.
>>>>>>>
>>>>>>>
>>>>>>
>>>>>> Did you notice what domain that Theorem was stated for?
>>>>>>
>>>>>> "First Order" deals with just a small limited subset of Mathematics.
>>>>>>
>>>>>> Yes, it is know that First Orders system can be complete, but can
>>>>>> not express all the known properties of the Natural Numbers.
>>>>>>
>>>>>> Comutation Theory is NOT a "First Order Predicate Calculus" (it
>>>>>> uses Higher Order logical predicated), so that Theorem doesn't apply.
>>>>>
>>>>> None-the-less its still applies to the philosophical foundation of
>>>>> every
>>>>> formal system. If an expression of formal or natural language is
>>>>> neither
>>>>> provable nor refutable then it is not a logical expression of any kind
>>>>> at all.
>>>>>
>>>>
>>>> Nope. Shows you don't understand what you are talking about,
>>>>
>>>> Note, it isn't that True is DEFINED as provable, but that for many
>>>> first order systems, it can be proven that all true statements can
>>>> be proven.
>>>>
>>>> Yes, it is possible to include in a system an Axiom/Definition that
>>>> says that all True Statements are Provable, and such a system needs
>>>> to limit its logical capability, like it can only use first order
>>>> predicates, or it will become inconsistent, and such inconsistancy
>>>> might not be detectable within the system.
>>>>
>>>> Such a system can NOT handle the fullness of Mathematics, or systems
>>>> like Computation Theory
>>>>
>>>
>>> It handles the subset of computations that are decision problems
>>> through the Curry Howard Isomorphism.
>>
>> No, Computation Theory uses logic operations not allowed
>> withinvsystems that meet the requirements of the Completeness Theorem.
>>
>> Curry Howard also only applies in systems that HAVE programs, which
>> aren't the sort of systems that the Completeness Theorem holds for.
>>
>> You are just showing you don't understand the concept of
>> pre-requirements of a Theory.
>>
>
> Every expression of language that cannot possibly resolve to true or
> false is not a logic expression. The computation deriving the square
> foot area of a house is not a logic expression.
>
>>>
>>>>> Everyone that has been debating me on this forum does so entirely
>>>>> on the
>>>>> basis of learn-by-rote dogma.
>>>>
>>>> Nope, you have been arguing from never-learned-due-to-stupdity dogma.
>>>>
>>>> Your IGNORANCE does make your statements true.
>>>>
>>>>>
>>>>> No one has considered that this math dogma might be inconsistent thus
>>>>> incoherent when examined through its foundation basis.
>>>>>
>>>>
>>>> No, you are just showing that you don't understanf what you are
>>>> talking about.
>>>
>>> The conventional definition of incompleteness:
>>> Incomplete(T) ↔ ∃φ ((T ⊬ φ) ∧ (T ⊬ ¬φ))
>>>
>>> is wrong the correct way to say this is
>>> Valid(φ) ↔ ((T ⊢ φ) ∨ (T ⊢ ¬φ))
>>>
>>
>> No, your "Valid" is just a restatement of Complete.
>>
>
> You are ignoring GIGO: garbage in, garbage out.
> Every self-contradictory expression of language that previously showed
> that a formal system is incomplete actually shows that the expression
> itself is garbage.

No, YOU are talking GIGO since the expresions that you are refering to
are NOT "Self-contradictory".

The statement that "Statement X is not provable" is NOT a
self-contradictory statement.

>
>> I don't know of a use of the Valid as applying to SYSTEMS, just
>> arguments/proofs. The closes to Validity is Consistent, which means
>> that no statement can be proven both True and False.
>>
>> There is nothing "Wrong" with an incomplete system that makes is not
>> valid.
>
> When the term "incomplete" is applied to formal systems in this way it
> is a God damned lie.

No, YOU are the "God Dammed Liar", because you don't speak the truth,
because it seems you don't actually KNOW the Truth.

Definitions are what they are.

Unless you have been lying about your health, you will likely find this
out at some point soon and make it empirical knowledge.

>
>>
>> That it does't meet your INCORRECT ideas, isn't grounds to claim they
>> are.
>