XPost: comp.theory, sci.math, comp.ai.philosophy   
   From: polcott333@gmail.com   
      
   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.   
      
   Inside PA, all you have are symbols and rules.   
   Whether a statement is “true” in the standard   
   model is something we say from outside PA.   
      
   That’s why Gödel can separate ‘true’ from   
   ‘provable’—because truth is defined externally.   
      
   If you stop using that external notion of truth   
   and rely only on the rules inside PA, then “true”   
   just means “provable,” and the incompleteness   
   gap disappears.   
      
      
   > Thus, the FACT that no number will statisfy the relationsip doesn't   
   > depend on anythign outside of that arithmatic.   
   >   
   > Thus, G is TRUE in the system, based on nothing but the basic rules of   
   > arithmatic in the system.   
   >   
   > It also turns out that there can not be a proof in the system, as even   
   > if the system can't understand the meaning if the statement in the meta-   
   > system, it still follows the results of that.   
   >   
   > You seem to think that the proof is based on some complicated logic that   
   > you can make not true. No, it is based on the fundamental properties of   
   > Mathematics, and the fact that Mathematics creates a truth-conditional   
   > system, even if you want to try to limit what you can understand of it.   
   >   
      
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