XPost: comp.theory, sci.logic, sci.math   
   From: polcott333@gmail.com   
      
   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.   
      
   --   
   Copyright 2026 Olcott

   
      
   My 28 year goal has been to make 
   
   "true on the basis of meaning expressed in language"
   
   reliably computable for the 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)