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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 | (8 earlier) <10kheqs$31ns8$1@dont-email.me> <1kYaR.41028$WtCb.6184@fx42.iad> <10khlr0$33k6m$1@dont-email.me> <L5ZaR.41030$WtCb.4931@fx42.iad> <10kho46$34969$1@dont-email.me> |
| From | Richard Damon <Richard@Damon-Family.org> |
| Message-ID | <aT8bR.115056$hu34.21850@fx46.iad> (permalink) |
| Organization | Forte - www.forteinc.com |
| Date | 2026-01-18 12:37 -0500 |
Cross-posted to 4 groups.
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. Unless you think that the sum of two numbers can change based on the meaning you have put into them, the assignment of meaning in the meta-ssytem didn't affact the results. Your problem is you are so stupid you can't understand that our knowledge of a system doesn't affect the truth in the system. Regardless of our understanding of the meta system, it is still a FACT that no number will ever satisfy the relationship, or that there can be no proof in the system of that fact. So, by MATH, there is no number that satisfies that relationship, and thus BY MATH the statement is true (but not determinable by a finite process in the system). Thus, all you are saying is that math itself is not-well-founded even though it is the results of basic logic and thus your logic is not-well-founded. The problem is that your attempt to use proof-theoretic interpretation of math just doesn't work.
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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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