Forums before death by AOL, social media and spammers... "We can't have nice things"
| comp.ai.philosophy | Perhaps we should ask SkyNet about this | 59,235 messages |
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| Message 59,100 of 59,235 |
| olcott to Richard Damon |
| =?UTF-8?Q?Re=3A_G=C3=B6del=27s_G_has_nev |
| 17 Jan 26 22:38:28 |
XPost: comp.theory, sci.logic, sci.math From: polcott333@gmail.com 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 -- Copyright 2026 Olcott |
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