XPost: comp.theory, sci.logic, sci.math   
   From: polcott333@gmail.com   
      
   On 1/17/2026 6:14 PM, Richard Damon wrote:   
   > On 1/17/26 5:50 PM, olcott wrote:   
   >> 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.   
   >   
   > But Truth *IS* Truth, or you are just misdefining it.   
   >   
   > The fact that a system can't tell you the truth value of a statement   
   > doesn't mean the statement doesn't have a truth value.   
   >   
   > And, the problem is that, as was shown, systems with a truth predicate   
   > CAN'T support PA or they are inconsistant.   
   >   
   > I guess systems that lie aren't a problem to you since you think lying   
   > is valid logic.   
   >   
   >>   
   >>>>   
   >>>> 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.   
   >   
   > No, he shows that any system that support PA and a Truth Predicate is   
   > inconstant.   
   >   
   > It seems you just want to let your system be inconsistent, as then you   
   > can "prove" whatever you want.   
   >   
   >>   
   >> PA can prove statements, but it cannot assert that   
   >> those statements are true. Those are different notions.   
   >   
   > Right, but statments in PA can be True even without such a predicate.   
   >   
      
   Unless PA can prove it then they never were actually   
   true in PA. They were true outside of PA in meta-math.   
      
      
      
      
   --   
   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)