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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 | (2 earlier) <10kh3ob$2ucs8$1@dont-email.me> <hBVaR.3767$Al3.1489@fx20.iad> <10kham4$30iju$1@dont-email.me> <EzWaR.3770$Al3.1707@fx20.iad> <10khd4b$319ho$1@dont-email.me> |
| From | Richard Damon <Richard@Damon-Family.org> |
| Message-ID | <TXWaR.3773$Al3.1879@fx20.iad> (permalink) |
| Organization | Forte - www.forteinc.com |
| Date | 2026-01-17 20:46 -0500 |
Cross-posted to 4 groups.
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? > > When you say 'truth goes beyond knowledge,' you're really saying 'meta- > theoretic truth goes beyond PA-provability.' That's trivially correct, > but it doesn't establish that Gödel sentences are 'true in arithmetic'— > it just shows they're true in a meta-theoretic framework we're smuggling > in and calling 'arithmetic.' Which just shows your confusion of Truth and Knowlege. And Godel's sentence is shown to be true in the system it was derived in. Not a "model" of that system, but in the basic system. > > The question isn't whether meta-theoretic truth transcends provability. > It's whether meta-theoretic truth has any claim to being genuinely > arithmetical truth, or whether it's a confusion of levels that's been > normalized for a century. > So, you don't think that 1 + 1 = 2 is a truth? I think you are just showing that you are trying to side step what truth actually is as you can't deal with it.
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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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