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Re: Gödel's G has never actually been true in arithmetic

Subject Re: Gödel's G has never actually been true in arithmetic
Newsgroups sci.logic, sci.math, comp.theory, comp.ai.philosophy
References (9 earlier) <10kjr33$3sepi$1@dont-email.me> <Pr3Az-UCqojeSktlkJZNevCeii0@jntp> <10kk7sq$75m$1@dont-email.me> <10kka23$3s3oo$1@dont-email.me> <10kkbsm$1fvj$1@dont-email.me>
From Richard Damon <Richard@Damon-Family.org>
Message-ID <7pEbR.400765$rbZb.175979@fx17.iad> (permalink)
Organization Forte - www.forteinc.com
Date 2026-01-20 00:29 -0500

Cross-posted to 4 groups.

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On 1/18/26 11:28 PM, olcott wrote:
> On 1/18/2026 9:56 PM, Richard Damon wrote:
>> On 1/18/26 10:19 PM, olcott wrote:
>>> On 1/18/2026 7:24 PM, Python wrote:
>>>> Le 19/01/2026 à 00:41, olcott a écrit :
>>>> ..
>>>>> I already just said that the proof and refutation of
>>>>> Goldbach are outside the scope of PA axioms.
>>>>>
>>>>> Any proof or refutation of Goldbach would have to use
>>>>> principles stronger than the axioms of PA, because PA
>>>>> itself does not currently derive either direction.
>>>>
>>>> "currently" ? ?  What kind of language is that? PA is what it is, it 
>>>> not changing with time !
>>>>
>>>> You could have said that about Fermat's theorem back in the day... 
>>>> It happens not to be the case.
>>>>
>>>> You are out of reason, Peter. Not only a liar, an hypocrite, but a 
>>>> fool.
>>>>
>>>
>>> If its truth value cannot be determined in a finite
>>> number of steps then it is not a truth bearer in PA,
>>> otherwise it is a truth-bearer in PA with an unknown value.
>>>
>>
>> So, you admit that you don't know how to classify it.
>>
>> Thus its truth-bearer status is unknown.
>>
>> Thus, your claim that it is outside of PA is just a LIE.
>>
> 
> No it was a mistake. Here is my correction:
> If Goldbach's truth value cannot be determined in a
> finite number of steps then it is not a truth bearer
> in PA, otherwise it is a truth-bearer in PA with an
> unknown truth value.

But,

> 
> This has no effect on my claim that I got rid of
> Gödel Incompleteness.

Sure it does, because it shows your system is not well founded.

> 
> When we change the foundation of formal systems
> to proof theoretic semantics and add my truth
> predicates then Gödel's claim of applying to
> every formal system that can do a little bit of
> arithmetic becomes simply false.
> 

But proof-theoretic semantics are not-well-founded when applied to 
systems like PA, as they need to use truth-conditional logic to 
determine their proof-theoretic fvalues.

> Every attempt at showing incompleteness <in> PA
> has never actually been <in> PA.

Sure they were in PA. PA as a system defines the basics of mathematics. 
It DEFINES a version of the Natural Numbers with a set of properties.

These properties can not all be resloved with the finite proofs that 
proof theoretic semantics allows.

In particular, you often can't determine that no proof exists (except by 
finding the proof of the negation of the statement) as there are an 
infinte number of possible proofs to rule out.

This means that actually PROVING that a statement is not-well-founded 
can't be done in a proof-theoretic manner.

> 
> The satisfaction of external models of arithmetic
> never has been <in> PA. These are categorically
> outside of PA by the definition of proof theoretic
> semantics thus defined as non-well-founded. This
> neuters their ability to show incompleteness.
> 

No, proof-theoretic semantics are just not well founded in PA.

As you can't determine a proof-theoretic truth value for some statements.

it isn't that the value is unknown, as that just means that further 
search can find the answer, but that literally there is NO valid 
proof-theoretic truth value by your definition.

There is no finite proof that it is true.
There is no finite proof that it is false.
There is no finite proof of the above two statements.

Thus, there is no proof-theoretic "truth value" for the statement, not 
even not-well-founded, so the definition creates a system that is not 
well founded.

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Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-17 15:08 -0600
  Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-17 16:54 -0500
    Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-17 16:50 -0600
      Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-17 19:14 -0500
        Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-17 18:49 -0600
          Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-17 20:20 -0500
            Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-17 19:30 -0600
              Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-17 20:46 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-17 19:59 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-17 22:20 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-17 21:59 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-17 23:13 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-17 22:38 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-18 12:37 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-18 12:38 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-18 15:55 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-18 15:49 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-18 18:28 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-18 17:41 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-18 19:28 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-18 21:17 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-20 00:29 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-20 10:50 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-20 23:00 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-18 21:19 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <news.x.richarddamon@xoxy.net> - 2026-01-18 22:56 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-18 22:28 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <news.x.richarddamon@xoxy.net> - 2026-01-19 06:49 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-19 08:43 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-20 00:29 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-20 15:23 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-20 23:04 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-20 22:54 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <news.x.richarddamon@xoxy.net> - 2026-01-21 07:35 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-21 09:45 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <news.x.richarddamon@xoxy.net> - 2026-01-21 22:37 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-21 21:53 -0600
                Re: Gödel's G has never actually been true in arithmetic Python <python@cccp.invalid> - 2026-01-22 04:59 +0000
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-21 23:18 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-22 19:17 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-22 18:33 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <news.x.richarddamon@xoxy.net> - 2026-01-22 21:51 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-22 22:18 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-23 20:33 -0500
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-22 19:15 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-19 13:20 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-20 00:29 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-20 14:00 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-20 23:12 -0500
                Re: Gödel's G has never actually been true in arithmetic Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2026-01-20 23:08 +0000
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-20 17:33 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-22 19:23 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-22 18:49 -0600
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-22 19:05 -0600
                Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-22 21:48 -0500
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-22 19:30 -0600
                Re: Gödel's G has never actually been true in arithmetic Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2026-01-23 00:23 +0000
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-22 18:29 -0600
                Re: Gödel's G has never actually been true in arithmetic Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2026-01-23 01:15 +0000
                Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-22 19:38 -0600
  Re: Gödel's G has never actually been true in arithmetic Mikko <mikko.levanto@iki.fi> - 2026-01-18 12:09 +0200
  Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-19 20:39 -0600
    Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-20 00:29 -0500
      Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-20 15:39 -0600
        Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-20 23:21 -0500
        Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-28 12:17 -0600
          Re: Gödel's G has never actually been true in arithmetic Richard Damon <news.x.richarddamon@xoxy.net> - 2026-02-01 07:33 -0500
      Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-28 12:08 -0600
        Re: Gödel's G has never actually been true in arithmetic Richard Damon <news.x.richarddamon@xoxy.net> - 2026-02-01 07:33 -0500

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