XPost: comp.theory, sci.logic, sci.math   
   From: polcott333@gmail.com   
      
   On 1/19/2026 11:29 PM, Richard Damon wrote:   
   > On 1/18/26 10:17 PM, olcott wrote:   
   >> 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   
      
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