XPost: comp.theory, sci.math, comp.ai.philosophy   
   From: polcott333@gmail.com   
      
   On 1/19/2026 11:29 PM, Richard Damon wrote:   
   > On 1/19/26 9:39 PM, olcott wrote:   
   >> 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))   
   >>   
   >>   
   >   
   > PA doesn't have a truth predicate, because it CAN'T.   
   >   
   > Tarski prove this, any actual truth predicate added to PA makes it   
   > inconsistant.   
   >   
   > The problem is your proof-theoretic statements need truth-conditional   
   > interpreation to be evaluated, as it is typically impossible to prove   
   > that no proof exists (except by proving the negation of the statement)   
   >   
   > This means that your logic is just based on not-well-founded ideas.   
      
   a meta‑level system is required to stand above PA and   
   filter expressions before PA ever evaluates them. The   
   meta‑system performs the structural work PA cannot do:   
   it detects cycles, blocks diagonalization, rejects   
   non‑truth‑bearers, and prevents PA from entering   
   infinite loops. Once the meta‑system has screened out   
   non‑well‑founded formulas, PA can safely apply its   
   internal truth predicate—defined purely as provability—to   
   the remaining expressions.   
      
      
   --   
   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   
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