XPost: sci.logic, sci.math, comp.theory
From: polcott333@gmail.com
On 1/25/2026 1:54 PM, Richard Damon wrote:
> On 1/25/26 2:09 PM, olcott wrote:
>> On 1/25/2026 12:36 PM, Richard Damon wrote:
>>> On 1/24/26 8:44 PM, olcott wrote:
>>>> On 1/24/2026 6:52 PM, Richard Damon wrote:
>>>>> On 1/24/26 5:31 PM, olcott wrote:
>>>>>> On 1/24/2026 4:25 PM, Richard Damon wrote:
>>>>>>> On 1/24/26 3:38 PM, olcott wrote:
>>>>>>>> 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.
>>>>>>>>
>>>>>>>
>>>>>>> Which means you don't understand what N actually is.
>>>>>>>
>>>>>>> Nothing can be "True in N" unless that truth comes from the
>>>>>>> Axioms of PA, as N is the result of PA.
>>>>>>>
>>>>>>
>>>>>> combined with the meta-math external model.
>>>>>
>>>>> Nope. N is just a set of object built in the Formal System defined
>>>>> by PA. 0 comes from Axiom 1 which states there is a 0.
>>>>>
>>>>
>>>> If G is true and not provable then you have
>>>> the wrong kind of true. I have known that
>>>> the entire body of knowledge is a semantic
>>>> tautology for 28 years.
>>>
>>> No, YOU do. The problem is Truth in the real world isn't based on
>>> being about to prove the fact, and most things are not actually
>>> provable, just well approximatable.
>>>
>>
>> That is why this insight was so important:
>> "true on the basis of meaning expressed in language"
>> I broke through the 75 year logjam of the analytic/synthetic
>> distinction.
>
> In other words, you don't accept the Pythgorean Theorem as "True", since
> its Tru-ness doesn't come out of the meaning of its words.
>
[meaning of its words]
My sentence is not restricted to words and
does include mathematical expressions.
--
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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