XPost: sci.logic, sci.math, comp.theory
From: polcott333@gmail.com
On 1/24/2026 1:52 PM, Richard Damon wrote:
> On 1/24/26 2:25 PM, olcott wrote:
>> On 1/24/2026 1:23 PM, Richard Damon wrote:
>>> On 1/24/26 12:54 PM, olcott wrote:
>>>> On 1/24/2026 11:10 AM, Richard Damon wrote:
>>>>> On 1/24/26 10:44 AM, olcott wrote:
>>>>>>
>>>>>> The statement that G is true and unprovable in PA has
>>>>>> always been counter-factual. It has never actually been
>>>>>> true PA and that is why it is unprovable in PA.
>>>>>
>>>>> Sure it is. At least it is a FACT that no natural number will
>>>>> statisfy that relationship, and there is no proof in PA of that fact.
>>>>>
>>>>
>>>> Have you ever heard of: "true in the standard model of arithmetic"?
>>>
>>>
>>> Sure, but they are not in Peano Arithmatic, but are (generally) 1st
>>> order variations of the Peano Axioms which lead to alternate number
>>> systems.
>>>
>>> Godel's proof is statd to be in a system with at least the properties
>>> of Peano Arithmatic, having the ability to show the properties of the
>>> "Natural Numbers"
>>>
>> Gödel’s incompleteness theorem only “works” if
>> one smuggles in an external notion of truth
>> (truth in ℕ) and then pretends it is an
>> internal notion of truth (truth in PA).
>> If we refuse to make that identification,
>> incompleteness evaporates.
>>
>
> But Truth in N is part of Peano Arithmatic, as Peano Arithmatic is a
> axiomiation to create the Natural Numbers.
>
You have that backwards. Truth in ℕ requires PA
as part of it, and PA itself has no notion of
Truth in ℕ. PA only has proofs from its own axioms
that can be construed as truth in PA, not truth in ℕ.
--
Copyright 2026 Olcott
My 28 year goal has been to make
"true on the basis of meaning expressed in language"
reliably computable for the entire body of knowledge.
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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