XPost: comp.theory, sci.logic, sci.math   
   From: polcott333@gmail.com   
      
   On 1/17/2026 7:20 PM, Richard Damon wrote:   
   > On 1/17/26 7:49 PM, olcott wrote:   
   >> 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.   
   >>   
   >   
   > Sure it is. Truth goes beyond knowledge.   
   >   
      
   You're assuming 'truth in arithmetic' means truth-in-the-standard-model.   
   But that's a meta-theoretic construct—it's truth about arithmetic from   
   outside PA, not truth in arithmetic. PA has no internal truth predicate   
   and no way to access the standard model from within.   
      
   When you say 'truth goes beyond knowledge,' you're really saying   
   'meta-theoretic truth goes beyond PA-provability.' That's trivially   
   correct, but it doesn't establish that Gödel sentences are 'true in   
   arithmetic'—it just shows they're true in a meta-theoretic framework   
   we're smuggling in and calling 'arithmetic.'   
      
   The question isn't whether meta-theoretic truth transcends provability.   
   It's whether meta-theoretic truth has any claim to being genuinely   
   arithmetical truth, or whether it's a confusion of levels that's been   
   normalized for a century.   
      
   --   
   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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