XPost: comp.theory, sci.logic, sci.math   
   From: polcott333@gmail.com   
      
   On 1/17/2026 3:08 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.   
   >   
   > 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.   
   >   
   > My work begins by correcting this foundational error.   
   >   
   > 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.   
   >   
      
   The only reason that anyone ever suggested an external measure of truth   
   as a proxy for actual truth  PA is because PA did not have its own   
   truth predicate. I fixed that anchored in PA's own axioms. Now we can   
   see that an external measure of true  PA was never actually true   
    PA at all. It was true about PA one level of indirect reference   
   away from true in PA. It was incorrectly conflated with true in PA   
   because no one saw any other alternatives.   
      
   ∀x ∈ PA ((True(PA, x)  ≡ (PA ⊢ x))   
   ∀x ∈ PA ((False(PA, x) ≡ (PA ⊢ ~x))   
   ∀x ∈ PA (~TruthBearer(PA, x) ≡ (~True(PA, x) ∧ (~False(PA, x))   
      
      
   --   
   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)