XPost: comp.theory, sci.logic, sci.math   
   From: polcott333@gmail.com   
      
   On 1/18/2026 5:28 PM, Richard Damon wrote:   
   > On 1/18/26 4:49 PM, olcott wrote:   
   >> 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   
      
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