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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 | <10kgto8$2s9mj$1@dont-email.me> <zyTaR.45932$ZE4d.15262@fx44.iad> <10kh3ob$2ucs8$1@dont-email.me> <hBVaR.3767$Al3.1489@fx20.iad> <10kham4$30iju$1@dont-email.me> |
| From | Richard Damon <Richard@Damon-Family.org> |
| Message-ID | <EzWaR.3770$Al3.1707@fx20.iad> (permalink) |
| Organization | Forte - www.forteinc.com |
| Date | 2026-01-17 20:20 -0500 |
Cross-posted to 4 groups.
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. Truth *IS* I guess your problem is you don't understand what truth really is. 1 + 1 = 2 IS true, and True in PA, doesn't matter that PA has no predicate that says so. Or, do you not understand how mathematics works.
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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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