XPost: comp.theory, sci.logic, sci.math   
   From: polcott333@gmail.com   
      
   On 1/17/2026 3:54 PM, Richard Damon wrote:   
   > On 1/17/26 4:08 PM, olcott wrote:   
   >> For nearly a century, discussions of arithmetic have quietly   
   >> relied on a fundamental conflation: the idea that   
   >> “true in arithmetic” meant “true in the standard model of ℕ.”   
   >> But PA itself has no truth predicate, no internal semantics,   
   >> and no mechanism for assigning truth values. So what was   
   >> called “true in arithmetic” was always meta-theoretic truth   
   >> about arithmetic, imported from an external model and never   
   >> grounded inside PA.   
   >   
   > Nope, just shows you don't understand what TRUTH means.   
   >   
      
   I’m distinguishing internal truth from external truth.   
   PA has no internal truth predicate, so it cannot express   
   or evaluate truth internally.   
      
   The only notion of truth available for PA is the external,   
   model‑theoretic one — which is meta‑theoretic by definition.   
      
   >>   
   >> This conflation was rarely acknowledged, and it shaped the   
   >> interpretation of Gödel’s incompleteness theorems, independence   
   >> results like Goodstein and Paris–Harrington, and the entire   
   >> discourse around “true but unprovable” statements.   
   >   
   > WHich Godel proves exsits.   
   >   
   >>   
   >> My work begins by correcting this foundational error.   
   >   
   > By LYING and destroying the meaninf of truth.   
   >   
   >>   
   >> PA has no internal truth predicate, so classical claims of   
   >> “true in arithmetic” were always meta-theoretic. My system   
   >> introduces a truth predicate whose meaning is anchored   
   >> entirely in PA’s axioms and inference rules, not in external   
   >> models. Any statement whose meaning requires meta-theoretic   
   >> interpretation or non-well-founded self-reference is rejected   
   >> as outside the domain of PA. This yields a coherent, internal   
   >> notion of truth in arithmetic for the first time.   
   >>   
   >   
   > Not having a "Predicate" doesn't mean not having a definition of truth.   
   >   
      
   A meta‑theoretic definition of truth is not the same   
   as an internal truth predicate. Tarski’s definition of   
   truth for arithmetic is external to PA and cannot be   
   expressed inside PA. That’s exactly the distinction   
   I’m drawing.   
      
   PA can prove statements, but it cannot assert that   
   those statements are true. Those are different notions.   
      
   > Your problem is that your "Truth Predicate" forces either you system to   
   > be "trivial" or "inconsistant".   
   >   
   > But, you are too stupid to understand this.   
   >   
   > Your own system requires that which you call non-well-founded, so it is   
   > itself, by your definition, not-well-founded.   
   >   
   > The problem is, except in trivial systems, we can't actually tell if a   
   > statement is well founded until we determine its truth, and may   
   > declerations of not-well-founded are themselves not-well-founded.   
   >   
   > You can only call Godel G statement not-well-founde by accept that it is   
   > true and unprovable, you can not otherwise PROVE that it isn't well-   
   > founded.   
      
      
   --   
   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)