XPost: comp.theory, sci.math   
   From: polcott333@gmail.com   
      
   On 1/26/2026 11:13 AM, Richard Damon wrote:   
   > On 1/26/26 11:58 AM, olcott wrote:   
   >> On 1/26/2026 10:45 AM, Richard Damon wrote:   
   >>> On 1/26/26 10:22 AM, olcott wrote:   
   >>>> On 1/26/2026 6:55 AM, Mikko wrote:   
   >>>>> On 25/01/2026 15:30, olcott wrote:   
   >>>>>> On 1/25/2026 5:24 AM, Mikko wrote:   
   >>>>>>> On 24/01/2026 16:18, olcott wrote:   
   >>>>>>>> On 1/24/2026 2:23 AM, Mikko wrote:   
   >>>>>>>>> On 22/01/2026 18:47, olcott wrote:   
   >>>>>>>>>> On 1/22/2026 2:21 AM, Mikko wrote:   
   >>>>>>>>>   
   >>>>>>>>>>> Anyway, what can be provven that way is true aboout PA. You   
   >>>>>>>>>>> can deny   
   >>>>>>>>>>> the proof but you cannot perform what is meta-provably   
   >>>>>>>>>>> impossible.   
   >>>>>>>>>   
   >>>>>>>>>> The meta-proof does not exist in the axioms of PA   
   >>>>>>>>>> and that is the reason why an external truth in   
   >>>>>>>>>> an external model cannot be proved internally in PA.   
   >>>>>>>>>> All of these years it was only a mere conflation   
   >>>>>>>>>> error.   
   >>>>>>>>>   
   >>>>>>>>> It is perfectly clear which is which. But every proof in PA is   
   >>>>>>>>> also   
   >>>>>>>>> a proof in Gödel's metatheory.   
   >>>>>>>>   
   >>>>>>>> ∀x ∈ PA (  True(PA, x) ≡ PA ⊢  x )   
   >>>>>>>> ∀x ∈ PA ( False(PA, x) ≡ PA ⊢ ¬x )   
   >>>>>>>> ∀x ∈ PA ( ¬WellFounded(PA, x) ≡   
   >>>>>>>>           (¬True(PA, x) ∧ (¬False(PA, x)))   
   >>>>>>>   
   >>>>>>> Those sentences don't mean anything without specificantions of a   
   >>>>>>> language and a theory that gives them some meaning.   
   >>>>>>   
   >>>>>> In other word you do not understand standard notational   
   >>>>>> conventions that define True for PA as provable from the   
   >>>>>> axioms of PA and False for PA as refutable from the axioms   
   >>>>>> of PA.   
   >>>>>   
   >>>>> There are no notational convention that defines True, False, and   
   >>>>> WellFounded with two arguments. And you did not specify in which   
   >>>>> context your sentences are true or otherwise relevant.   
   >>>>>   
   >>>>   
   >>>> “x is a single finite string representing   
   >>>> a PA‑formula, such as ‘2 + 3 = 5’.   
   >>>> True(PA, x), False(PA, x), and WellFounded(PA, x)   
   >>>> are meta‑level unary predicates classifying   
   >>>> that formula by its provability in PA.”   
   >>>>   
   >>>   
   >>> In outher words, you ACCEPT that the meta level can define what is   
   >>> true in PA?   
   >>>   
   >>> I thought you said that PA had to be able to determine the truth itself?   
   >>   
   >> We need a meta-level truth predicate anchored   
   >> only in the axioms of PA itself and thus not   
   >> anchored in the standard model of arithmetic.   
   >>   
   >   
   > But Peano Arithmatic *IS* a standard model of arithmetic.   
   >   
   > The Induction Axiom makes it so.   
   >   
      
   PA is a first‑order theory, not a model. The induction   
   axiom does not force the standard model; PA has many   
   non‑standard models. My meta‑level truth predicate   
   is proof‑theoretic: True(PA, x) means PA ⊢ x. It does   
   not appeal to truth in the standard model of arithmetic.   
      
   > The other models tend to come from making a variant of PA by changing   
   > that 2nd order Induction Axiom to various first order versions to   
   > "simulate" its power using the rest of the Peano Axioms, and then adding   
   > something more to complete it.   
      
      
   --   
   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   
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