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Groups > comp.theory > #139282
| From | olcott <polcott333@gmail.com> |
|---|---|
| Newsgroups | comp.theory, sci.logic, sci.math, comp.ai.philosophy |
| Subject | Re: Gödel's G has never actually been true in arithmetic |
| Date | 2026-01-20 10:50 -0600 |
| Organization | A noiseless patient Spider |
| Message-ID | <10kobp3$1cohe$1@dont-email.me> (permalink) |
| References | (17 earlier) <90ebR.121312$UIC2.42256@fx11.iad> <10kjr33$3sepi$1@dont-email.me> <wUebR.122034$UIC2.119582@fx11.iad> <10kk7of$71m$1@dont-email.me> <5pEbR.400761$rbZb.224761@fx17.iad> |
Cross-posted to 4 groups.
On 1/19/2026 11:29 PM, Richard Damon wrote: > On 1/18/26 10:17 PM, olcott wrote: >> On 1/18/2026 6:28 PM, Richard Damon wrote: >>> On 1/18/26 6:41 PM, olcott wrote: >>>> On 1/18/2026 5:28 PM, Richard Damon wrote: >>>>> On 1/18/26 4:49 PM, olcott wrote: >>>>>> On 1/18/2026 2:55 PM, Richard Damon wrote: >>>>>>> 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. >>>>>>> >>>>>> >>>>>> True(PA, x) ≡ PA ⊢ x >>>>>> does not require PA to search all proofs. It simply states: >>>>>> ---If PA proves x, then True(PA, x) holds. >>>>>> ---If PA does not prove x, then True(PA, x) does not hold. >>>>> >>>>> And how can you tell if PA proves something? >>>>> >>>> >>>> Every expression such as "2 + 3 = 5" that can be verified >>>> entirely on the basis of PA axioms is provable in PA. >>>> >>>>> You might know of the proof, but there might be one you don't know. >>>>> >>>>> THus, you STILL need a state for Truth Value exists but is unknown. >>>>> >>>>>> >>>>>>> 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. >>>>>> >>>>>> Goldbach is outside PA because PA neither proves >>>>>> it nor refutes it. In a proof‑theoretic framework, >>>>>> a statement belongs to PA’s inferential domain only >>>>>> if it is derivable from PA’s axioms. Since Goldbach >>>>>> is undecidable in PA, it has no inferential grounding >>>>>> there. Therefore, if a proof of Goldbach exists at >>>>>> all, it must lie outside PA’s deductive power. >>>>>> >>>>> >>>>> DO you KNOW that PA can't prove it? or is it you just don't know of >>>>> a way to prove it in PA. >>>>> >>>>> Do you KNOW that PA can't refute it? or is it you just haven't >>>>> found a refuation. >>>>> >>>>> If you can actually prove one of those statement then you will be >>>>> famous. >>>>> >>>> >>>> I already just said that the proof and refutation of >>>> Goldbach are outside the scope of PA axioms. >>> >>> But you didn't PROVE it, you just claim it based on lack of knowledge. >>> >>>> >>>> Any proof or refutation of Goldbach would have to use >>>> principles stronger than the axioms of PA, because PA >>>> itself does not currently derive either direction. >>> >>> Why do you say that? >>> >>> Can you PROVE it? >>> >> >> If its truth value cannot be determined in a finite >> number of steps then it is not a truth bearer in PA, >> otherwise it is a truth-bearer in PA with an unknown value. >> >> > > And how do you determine if its truth value cannot be determined in a > finite number of steps? > > Your proof-theoretic definitions still require truth-conditional logic > to be used. > > The bigger problem is that we have statements that can not be shown by > proof-theoretic means to be one of True, False, or Not-Well-Founded, and > in fact forcing that makes a contradiction. > PA is only a little tiny example of how my greater system that makes the body of knowledge that is "true on the basis of meaning expressed in language" computable. Unknowns are outside of this domain. > For instance, look at Godel's G, which states that there is no natural > number g that satisfies a given computable relationship, which is a pure > mathematical operation, so thus totally determined. > You know that it was never a pure mathematical operation it is all performed in meta-math. > This statement can NOT be proven to be not-well-founded, as to do so > means we can prove that its converse isn't true. which means we can > prove that no number g can exist that meets the requirement, (as if it > could, we couldn't prove that the statement can't be false), and thus we > now HAVE a proof that it is true, as that condition is EXACTLY what the > statement claims. > > The problem comes because some problem are inhenently following the laws > of the excluded middle, and thus MUST be True or False. But while they > must be true or false, there doesn't need to be a finite proof that > makes that true. > Two levels of the law of the excluded middle are required Is X a truth-bearer if yes then is x true > Godel's statement is an example of this, as mathementics, because of it > correlation to programming, is able to create "computations" that embue > meaning into the numbers, a meaning that can't be seen in the base > number system, but is "understood" by the program/relationship that was > created with it. This means that while PA doesn't understand the > meaning, the determinism of mathematics brings the results of that > meaning into the system. > > The relationship turns out to be a proof checker, in particular, a proof > checker for the statement of G. a number represents a "statement" (or > seires of statements) in PA, and a number that satisfies it will > represent a valid proof of G. > > Thus, if a number existed, then the statements it represented exist in > PA, and those statements become a proof that no such number could exist. > > Since this is logically impossible it can not be, thus the statement > must be true. > > But, if a proof existed of this fact, then we could compute in the meta > system the number that proof represents, and that number would by the > construciton of the relationship statisfy the relationship, makeing the > statement false. > > So, unless you think that it is possible for there to be a proof that a > false statement is true, the statement MUST be true but unprovable. > > That, or you get crasyness like mathematics is inconsistant, that some > basic mathematical operation of two natural numbers can give different > results at different times. As in while we THINK that 1 + 2 = 3, it > might be that sometimes 1 + 2 = 4. > > That, or you think that it is impossible to create a program that given > a proof in a specified system, checks that the proof is valid with 100% > certainty in that system. > > Sorry, your problem is that your concept just can't work in PA and > similar systems. > > This is one way to interprete Godel's Incompleteness proof. Gödel’s G is not a truth‑bearer inside PA. It is only interpretable as true in an external semantic model. The classical argument that G is “true but unprovable” relies entirely on meta‑mathematical assumptions about ℕ, satisfaction, and semantic bivalence. Once truth is internalized, the argument no longer applies. G is simply outside the domain of PA’s internal truth predicate. Therefore the classical Gödel conclusion is not a fact about PA, but a fact about how the meta‑theory interprets PA. -- Copyright 2026 Olcott<br><br> My 28 year goal has been to make <br> "true on the basis of meaning expressed in language"<br> reliably computable.<br><br> This required establishing a new foundation<br>
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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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