XPost: comp.theory, sci.logic, sci.math   
   From: polcott333@gmail.com   
      
   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.   
   >>>   
   >>> No, it uses just the math of PA.   
   >>>   
   >>> The meta-system just embues some additional meaning into the numbers.   
   >>>   
   >>   
   >> That is where it steps outside of math   
   >   
   > But that meaning doesn't actually affect the results in the system, only   
   > to let us KNOW the results.   
   >   
      
   ∀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))   
      
      
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