XPost: comp.theory, sci.math, comp.ai.philosophy   
   From: polcott333@gmail.com   
      
   On 1/18/2026 6:28 PM, Richard Damon wrote:   
   > On 1/18/26 6:41 PM, olcott wrote:   
   >> 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?   
   >>>>>>>>>>>>>   
      
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