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Groups > comp.theory > #139352
| From | Richard Damon <news.x.richarddamon@xoxy.net> |
|---|---|
| Newsgroups | sci.math, sci.logic, comp.theory |
| Subject | Re: Gödel's G has never actually been true in arithmetic |
| Date | 2026-01-22 21:51 -0500 |
| Organization | A noiseless patient Spider |
| Message-ID | <10kunnd$306qt$3@dont-email.me> (permalink) |
| References | (9 earlier) <10ks70p$2m8fa$1@dont-email.me> <-M8w7ApLHAiK9B-bPjcdK_CB6vQ@jntp> <10ksbuu$2nled$1@dont-email.me> <56zcR.306465$UIC2.259537@fx11.iad> <10kufka$3g34e$1@dont-email.me> |
Cross-posted to 3 groups.
On 1/22/26 7:33 PM, olcott wrote: > On 1/22/2026 6:17 PM, Richard Damon wrote: >> On 1/22/26 12:18 AM, olcott wrote: >>> On 1/21/2026 10:59 PM, Python wrote: >>>> Le 22/01/2026 à 04:54, olcott a écrit : >>>>> On 1/21/2026 9:37 PM, Richard Damon wrote: >>>>>> On 1/21/26 10:45 AM, olcott wrote: >>>>>>> On 1/21/2026 6:35 AM, Richard Damon wrote: >>>>>>>> On 1/20/26 11:54 PM, olcott wrote: >>>>>>>>> On 1/20/2026 10:04 PM, Richard Damon wrote: >>>>>>>>>> On 1/20/26 4:23 PM, olcott wrote: >>>>>>>>>>> On 1/19/2026 11:29 PM, Richard Damon wrote: >>>>>>>>>>>>> My system is not supposed to decide in advance whether >>>>>>>>>>>>> Goldbach is well‑founded. A formula becomes a truth‑bearer >>>>>>>>>>>>> only when PA can classify it in finitely many steps. >>>>>>>>>>>>> Goldbach may or may not be classifiable; that’s an open >>>>>>>>>>>>> computational fact, not a semantic requirement. This has >>>>>>>>>>>>> no effect on Gödel, because Gödel’s sentence is structurally >>>>>>>>>>>>> non‑truth‑bearing, not merely unclassified. >>>>>>>>>>>> >>>>>>>>>>>> Which shows that you don't understand what logic systems are. >>>>>>>>>>>> >>>>>>>>>>>> The don't "Decide" on truths, they DETERMINE what is true. >>>>>>>>>>>> >>>>>>>>>>>> Your problem is that either there is, or there isn't a >>>>>>>>>>>> finite length proof of the statement. >>>>>>>>>>>> >>>>>>>>>>>> Semantics can't change in a formal system, or they aren't >>>>>>>>>>>> really semantics. >>>>>>>>>>>> >>>>>>>>>>>> Your problem is you don't understand Godel statement, as it >>>>>>>>>>>> *IS* truth bearing as it is a simple statement with no >>>>>>>>>>>> middle ground, does a number exist that satisfies a given >>>>>>>>>>>> relationship. Either there is, or there isn't. No other >>>>>>>>>>>> possiblity. >>>>>>>>>>>> >>>>>>>>>>>> You confuse yourself by forgetting that words have actual >>>>>>>>>>>> meaning, and that meaning can depend on using the right >>>>>>>>>>>> context. >>>>>>>>>>>> >>>>>>>>>>>> Godel's G is a statement in the system PA. >>>>>>>>>>>> >>>>>>>>>>>> It is a statement about the non-existance of a natural >>>>>>>>>>>> number that satisfies a particular computable realtionship. >>>>>>>>>>>> >>>>>>>>>>>> It is a statement defined purely by mathematics and thus >>>>>>>>>>>> doesn't "depend" on other meaning. >>>>>>>>>>>> >>>>>>>>>>>> It is a mathematical FACT, that for this relationship, no >>>>>>>>>>>> matter what natural number we test, none will satisfy it, so >>>>>>>>>>>> its assertation that no number satisfies it makes it true. >>>>>>>>>>> >>>>>>>>>>> PA augmented with its own True(PA,x) and False(PA,x) >>>>>>>>>>> is a decider for Domain of every expression grounded >>>>>>>>>>> in the axioms of PA. >>>>>>>>>> >>>>>>>>>> No, it becomes inconsistant. >>>>>>>>>> >>>>>>>>>>> >>>>>>>>>>> A system at a higher level of inference than PA can >>>>>>>>>>> reject any expressions that define a cycle in the >>>>>>>>>>> directed graph of the evaluation sequence of PA >>>>>>>>>>> expressions. Then PA could test back chained inference >>>>>>>>>>> from expression x and ~x to the axioms of PA. >>>>>>>>>>> >>>>>>>>>> >>>>>>>>>> But there is no "cycle" in the statement of G. It is PURELY a >>>>>>>>>> statement of the non-existance of a number that satisfies a >>>>>>>>>> purely mathematic relationship (which has no meaning by itself >>>>>>>>>> in PA). >>>>>>>>>> >>>>>>>>> >>>>>>>>> Even the relationship cannot exist <in> PA. >>>>>>>>> Instead it is about PA in outside model theory >>>>>>>> >>>>>>>> No, it doesn't mention PA, it is about the numbers that are IN PA. >>>>>>>> >>>>>>>> Your problem is you forget to actually know what Godel's G is, a >>>>>>>> you only read the Reader's Digest version of the proof, as that >>>>>>>> is all you can understand. >>>>>>>> >>>>>>>> That, or you are saying that mathematics itself isn't in PA, and >>>>>>>> that you proof-theoretic stuff isn't in PA either, >>>>>>>> >>>>>>>> Sorry, you are just showing how ignorant you are. >>>>>>>> >>>>>>> >>>>>>> G_F ↔ ¬Prove_F(Gödel_Number(G_F)) contains a semantic >>>>>>> dependency loop, because evaluating G_F requires >>>>>>> evaluating Prove_F on the Gödel number of G_F, which >>>>>>> in turn requires evaluating G_F again; >>>>>> >>>>>> But that isn't G_F >>>>>> >>>>>> G_F is a statement that a particular relationship (lets call it >>>>>> R(x) ) will not be satisfied for any natural number x. >>>>>> >>>>> >>>>> That relationship has never existed inside actual >>>>> arithmetic >>>> >>>> It actually IS a relationship in the domain of PA. PUNTO. >>>> >>>> It is what it is. Denial is hopeless. >>> >>> When PA is actually given its own truth predicate >>> anchored only in its own axioms then for the first >>> time one see that meta-math truth in the standard >>> model of arithmetic never was actually true in PA >>> itself at all. >>> >> >> But PA can't be given such a truth predicate and reamin consistant. >> > > Unless the foundation model theory is replaced > with the foundation of proof theory and proof > theory itself is grounded in Haskell Curry's > notion of "true in the system". Try to show that working, and HAVE a truth predicate. Remember, a truth predicate is "True" if the input is a true expression, and "False" if the input is something else, being either a False statement, or a not-well-founded statement, or even just plain non-sense. > >> Your provblem is you are too stupid to understand the problem. >> >> I guess you claim is that if the meta arithmatic uses the fact that 2 >> * 3 = 6, then maybe in the base arithmatic 2 * 3 might now be 8. > >
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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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