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| Subject | Re: Gödel's G has never actually been true in arithmetic |
|---|---|
| Newsgroups | comp.theory, sci.logic, sci.math, comp.ai.philosophy |
| References | (10 earlier) <10khlr0$33k6m$1@dont-email.me> <L5ZaR.41030$WtCb.4931@fx42.iad> <10kho46$34969$1@dont-email.me> <aT8bR.115056$hu34.21850@fx46.iad> <10kj9bu$3m4vd$1@dont-email.me> |
| From | Richard Damon <Richard@Damon-Family.org> |
| Message-ID | <mNbbR.5234$px9c.1292@fx37.iad> (permalink) |
| Organization | Forte - www.forteinc.com |
| Date | 2026-01-18 15:55 -0500 |
Cross-posted to 4 groups.
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 >>>>>>> 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)) > > When we look at what is actually true directly in PA > and not what is true about PA in meta-math then Gödel > Incompleteness cannot arise. The nearly century long > mistake was conflating true about PA in meta-math for > what is actually true in PA. > > Except that none of the those statements are well-formed for all x, since we can't check ALL possible proofs (since there is an infinite number of them) to determine if a given statement is True, False, or Not a TruthBearer. You criteria only works in a system with only a finite number of possible proofs, of which PA doesn't fit. For instance, Which is the Goldbach conjecture? We think it is likely true, but don't have a proof YET. There COULD be a counter example, but we haven't found it. It might not be provable, but we don't know that either. Thus, your system can't even classify a simple problem, because your criteria are not well-founded.
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Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-17 15:08 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-17 16:54 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-17 16:50 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-17 19:14 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-17 18:49 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-17 20:20 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-17 19:30 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-17 20:46 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-17 19:59 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-17 22:20 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-17 21:59 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-17 23:13 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-17 22:38 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-18 12:37 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-18 12:38 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-18 15:55 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-18 15:49 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-18 18:28 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-18 17:41 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-18 19:28 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-18 21:17 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-20 00:29 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-20 10:50 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-20 23:00 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-18 21:19 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <news.x.richarddamon@xoxy.net> - 2026-01-18 22:56 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-18 22:28 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <news.x.richarddamon@xoxy.net> - 2026-01-19 06:49 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-19 08:43 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-20 00:29 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-20 15:23 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-20 23:04 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-20 22:54 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <news.x.richarddamon@xoxy.net> - 2026-01-21 07:35 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-21 09:45 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <news.x.richarddamon@xoxy.net> - 2026-01-21 22:37 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-21 21:53 -0600
Re: Gödel's G has never actually been true in arithmetic Python <python@cccp.invalid> - 2026-01-22 04:59 +0000
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-21 23:18 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-22 19:17 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-22 18:33 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <news.x.richarddamon@xoxy.net> - 2026-01-22 21:51 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-22 22:18 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-23 20:33 -0500
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-22 19:15 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-19 13:20 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-20 00:29 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-20 14:00 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-20 23:12 -0500
Re: Gödel's G has never actually been true in arithmetic Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2026-01-20 23:08 +0000
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-20 17:33 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-22 19:23 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-22 18:49 -0600
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-22 19:05 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-22 21:48 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-22 19:30 -0600
Re: Gödel's G has never actually been true in arithmetic Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2026-01-23 00:23 +0000
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-22 18:29 -0600
Re: Gödel's G has never actually been true in arithmetic Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2026-01-23 01:15 +0000
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-22 19:38 -0600
Re: Gödel's G has never actually been true in arithmetic Mikko <mikko.levanto@iki.fi> - 2026-01-18 12:09 +0200
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-19 20:39 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-20 00:29 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-20 15:39 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <Richard@Damon-Family.org> - 2026-01-20 23:21 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-28 12:17 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <news.x.richarddamon@xoxy.net> - 2026-02-01 07:33 -0500
Re: Gödel's G has never actually been true in arithmetic olcott <polcott333@gmail.com> - 2026-01-28 12:08 -0600
Re: Gödel's G has never actually been true in arithmetic Richard Damon <news.x.richarddamon@xoxy.net> - 2026-02-01 07:33 -0500
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