XPost: comp.theory, sci.logic, sci.math   
   From: polcott333@gmail.com   
      
   On 1/18/2026 2:55 PM, Richard Damon wrote:   
   > On 1/18/26 1:38 PM, olcott wrote:   
   >> On 1/18/2026 11:37 AM, Richard Damon wrote:   
   >>> On 1/17/26 11:38 PM, olcott wrote:   
   >>>> On 1/17/2026 10:13 PM, Richard Damon wrote:   
   >>>>> On 1/17/26 10:59 PM, olcott wrote:   
   >>>>>> On 1/17/2026 9:20 PM, Richard Damon wrote:   
   >>>>>>> On 1/17/26 8:59 PM, olcott wrote:   
   >>>>>>>> On 1/17/2026 7:46 PM, Richard Damon wrote:   
   >>>>>>>>> On 1/17/26 8:30 PM, olcott wrote:   
   >>>>>>>>>> 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.   
   >>>>>>>>>   
   >>>>>>>>> No, PA (Peano Arithmetic) itself defines the numbers and the   
   >>>>>>>>> arithmatic.   
   >>>>>>>>>   
   >>>>>>>>> Why do you think otherwise?   
   >>>>>>>>>   
   >>>>>>>>> And why does it NEED to access the model from within?   
   >>>>>>>>>   
   >>>>>>>>   
   >>>>>>>> Gödel‑style incompleteness only appears when “truth” is   
   >>>>>>>> defined using an outside model of the natural numbers.   
   >>>>>>>   
   >>>>>>> No, it uses the innate properties of the Natural Nubmers.   
   >>>>>>>   
   >>>>>>   
   >>>>>> meta-math is outside of math.   
   >>>>>>   
   >>>>>>>>   
   >>>>>>>> If you stop using model‑theoretic truth and rely only   
   >>>>>>>> on the meanings that come from the rules of the system   
   >>>>>>>> itself, then “true” and “provable” coincide — so the   
   >>>>>>>> incompleteness gap never arises.   
   >>>>>>>   
   >>>>>>> That doesn't make sense. The answer to the arithmatic doesn't   
   >>>>>>> depend on anything outside the rules, as numbers mean themselves.   
   >>>>>>>   
   >>>>>>> That a number statisfies the relationship derived doesn't depend   
   >>>>>>> on anything outside of that arithmatic.   
   >>>>>>>   
   >>>>>>   
   >>>>>> meta-math is outside of math.   
      
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