XPost: sci.math, comp.theory, comp.ai.philosophy
From: polcott333@gmail.com
On 1/19/2026 5:49 AM, Richard Damon wrote:
> 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.
>>
>> This has no effect on my claim that I got rid of
>> Gödel Incompleteness.
>
> Sure it does. As your system is just not well founded by its own
> definitios,
>
My system is not supposed to decide in advance whether
Goldbach is well‑founded. A formula becomes a truth‑bearer
only when PA can classify it in finitely many steps.
Goldbach may or may not be classifiable; that’s an open
computational fact, not a semantic requirement. This has
no effect on Gödel, because Gödel’s sentence is structurally
non‑truth‑bearing, not merely unclassified.
>>
>> 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 you CAN'T do that and keep the systems.
>
I am keeping the systems.
I’m changing the semantics.
PA’s syntax, axioms, and rules stay exactly
the same. What changes is that truth is internal
- finite derivability - and external model‑theoretic
satisfaction is no longer imported into PA. Gödel’s
claim depends on that external semantics, so once it’s
removed, his universal claim simply doesn’t apply.
>>
>> Every attempt at showing incompleteness PA
>> has never actually been PA.
>
> Sure it is.
>
> Godel's G shows your system is not well founded.
>
Gödel’s G only “shows” anything if you assume
classical semantic truth in an external model.
My system does not use that semantics. Truth in
PA is finite derivability; anything PA cannot
classify is not a truth‑bearer.
Gödel’s G is therefore not a truth‑bearer, not
a counterexample, and not evidence of ill‑foundedness.
You’re evaluating my system using assumptions it
does not adopt.
>>
>> The satisfaction of external models of arithmetic
>> never has been 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.
>>
>>
>
> But you system is just non-well-founded in PA.
>
> Godel's G has NO truth value, not even non-well-founded in PA by your
> system, and thus your system is broken.
>
> The problem is that for statements like it that have the property of not
> being having a known truth value if not provable, you system just breaks
> down.
>
> There is no proof of it being true, so it can't be true.
> There is no proof of it being false, so it can't be false.
> There is no proof of being not-well-founded, so it can't be non-well-
> founded.
>
> Your classification of claiming it to be non-well-founded is just non-
> well-founded.
>
> In fact, by your systems definitions, the claim of it being non-well-
> founded is non-well-founded as we can't prove it to be non-well-founded,
> as if it WAS not-well-founded, that means that you were able to prove
> that there wasn't a proof of it being false, which means there can't be
> a number that satisfies the requirement, as any number that existed
> forms an easy proof of falsehood, and thus must be true.
>
> So, there CAN'T be a proof of it not being well-founded.
>
> But if it isn't not-well-founded, then by your definition it must be
> True or False, which you already said it couldn't be.
>
> THus the only choice left is it not-well-founded that it is not-well-
> founded.
>
> But that arguement extends for that statement, so it is not-well-founded
> that the not-well-foundedness of the stsatement is not-well-founded.
>
> Thus, your system breaks with an infinite progression of not being able
> to classify the truth of the statement.
>
> So, the reason you think that Godel's (are related) proofs aren't well
> founded in PA is that your system is just not-well-founded in PA, but
> refuse to accept it,
>
> The problem is that definition of Truth is just incompatible with PA,
> which is why it can't be used.
>
> The problem is that the system has become "complex" enough that it
> inherently has grown bigger than provability of all things in it, and
> thus the concept of Truth being based on Provability just breaks as it
> means some things have undefinable (not just unknowable) truth values,
> they can't even be defined as not-having a truth value, as you can't
> prove that, but you insist that truth must be provable.
>
>
G is a truth bearer outside of PA in meta-math in the
same way that the Liar Paradox becomes true when it
refers to a different instance of itself.
This sentence is not true: "This sentence is not true"
is true because the inner sentence is not a truth bearer.
--
Copyright 2026 Olcott
My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable.
This required establishing a new foundation
--- SoupGate-Win32 v1.05
* Origin: you cannot sedate... all the things you hate (1:229/2)
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