XPost: sci.math, comp.theory, comp.ai.philosophy   
   From: polcott333@gmail.com   
      
   On 1/19/2026 11:29 PM, 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.   
   >   
   > 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.   
   >   
      
   Not at all. At you own repeated insistence the   
   domain of all of my systems is the set of knowledge   
   "true on the basis of meaning expressed in language"   
      
   >>   
   >> 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.   
   >   
      
   “You’re assuming proof‑theoretic semantics must be grounded   
   in truth‑conditional semantics. That assumption is false.   
   In proof‑theoretic semantics, meaning is given by inferential   
   rules, not external truth‑conditions.   
      
   So the internal truth predicate for PA is perfectly well‑founded,   
   and Gödel’s semantic argument no longer applies.”   
      
   >> Every attempt at showing incompleteness  PA   
   >> has never actually been  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 only reason anyone ever treated an external, model‑theoretic   
   notion of truth as a proxy for truth in PA is that PA originally   
   lacked its own internal truth predicate.   
      
   Once you anchor a truth predicate directly in PA’s axioms, it   
   becomes clear that the so‑called ‘truth in PA’ used by Gödel and   
   Tarski was never truth in PA at all.   
      
   It was truth about PA — one level of meta‑mathematical reference   
   removed. The two were conflated only because no one had a viable   
   alternative at the time.”   
      
   >>   
   >> 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.   
   >>   
   >   
   > 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.   
   >   
   >   
      
   In my system of PA non-well founded x can always be   
   detected one of two ways within the body of knowledge   
   that can be expressed as language.   
      
   There is no finite back chained inference from   
   x or ~x to the axioms of PA. The inference that   
   does exist has a cycle in the directed graph of   
   its evaluation sequence.   
      
      
      
      
      
      
   --   
   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)