Path: csiph.com!eternal-september.org!feeder.eternal-september.org!reader01.eternal-september.org!.POSTED!not-for-mail From: Keith Thompson Newsgroups: comp.theory,comp.ai.philosophy,comp.ai.nat-lang,sci.lang.semantics Subject: Re: Simply defining =?utf-8?Q?G=C3=B6del?= Incompleteness and Tarski Undefinability away V24 (Are we there yet?) Date: Thu, 16 Jul 2020 11:46:51 -0700 Organization: None to speak of Lines: 145 Message-ID: <87sgdrz49w.fsf@nosuchdomain.example.com> References: <2tCdnb0urbddzpfCnZ2dnUU7-b_NnZ2d@giganews.com> <87k0z85tt0.fsf@nosuchdomain.example.com> <87d0505kmk.fsf@nosuchdomain.example.com> <5Lmdnehh4P6hLZbCnZ2dnUU7-LdQAAAA@giganews.com> <878sfo5elp.fsf@nosuchdomain.example.com> <87zh820x98.fsf@nosuchdomain.example.com> <87imeo1wov.fsf@nosuchdomain.example.com> <87a7001bhr.fsf@nosuchdomain.example.com> Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: 8bit Injection-Info: reader02.eternal-september.org; posting-host="02ec520d1b2b3210c7ee6ae74092840a"; logging-data="6653"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX19+Yaaal2E0UcAPDZ5VsHX5" User-Agent: Gnus/5.13 (Gnus v5.13) Emacs/26.3 (gnu/linux) Cancel-Lock: sha1:GI3TPXxdKbEsOL4iD0v1lfdNFTo= sha1:3wNfRlKHZ5KBDolYWIF/wopKGuA= Xref: csiph.com comp.theory:21706 comp.ai.philosophy:22054 comp.ai.nat-lang:2432 olcott writes: > On 7/15/2020 8:42 PM, Keith Thompson wrote: >> olcott writes: >>> On 7/15/2020 1:04 PM, Keith Thompson wrote: >>>> olcott writes: >>>>> On 7/14/2020 1:25 PM, Keith Thompson wrote: >>>>>> olcott writes: >>>>>> [...] >>>>>>> Since everyone here is indoctrinated into believing that Gödel is >>>>>>> correct I have to use different terms for provability so that people >>>>>>> will carefully analyze my reasoning and not simply dismiss it >>>>>>> out-of-hand on the basis of their indoctrination. >>>>>> >>>>>> It seems to me that the best way to demonstrate that Gödel is >>>>>> incorrect would be to demonstrate a flaw in what he actually wrote. >>>>>> I haven't read everything you've written here, but I don't recall >>>>>> you ever directly quoting Gödel's proof. >>>>> >>>>> Not really. When we refute the enormously simplified key result of his >>>>> claim: true and unprovable can possibly coexist, then the steps that >>>>> he used to get to this key result are moot. >>>> >>>> You've been asserting that for years, and nobody believes you >>> https://scholar.google.com/scholar?hl=en&as_sdt=0%2C28&q=%22true+and+unprovable%22+godel&btnG=&oq=%22true+and+unprovable%22 >> >> 125 results. No, I'm not going to read them. > > 125 different people that all believe that Gödel showed that true and > unprovable formulas exists, and 125 > 0, thus "nobody believes you" is > proven to be false. Wait, what? Is that really what you meant to say? Gödel *did* show that true and unprovable formulas exist. Did you omit a "not"? OK, "nobody believes you" was hyperbole. I've seen nobody posting here who believes that you've successfully refuted Gödel's proof. If you can cite an exception, I suppose it would be mildly interesting, but not particularly relevant other than to refute my statement. I'll gladly revise it to "Hardly anybody believes you". >>>> Do you think that's going to change if you assert it just one >>>> more time? What is your goal here? >>>> >>>> Whether it's the best way or not, surely *a* way to demonstrate >>>> that Gödel is incorrect would be to demonstrate a flaw in what he >>>> actually wrote. Not in some summary of his proof, but in his actual >>>> proof as he wrote it. Something like "In step 42, Gödel makes use >>>> of this assumption, but previously in step 23 he showed that that >>>> assumption does not hold in all cases". (That's a hypothetical >>>> example, of course.) >>> >>> His mistake can only be seen through a refutation of the essence of >>> his conclusion. >> >> Ah, now that's an interesting assertion. Did you really mean "only"? > > "Needle in a hay stack" > When you are looking for a particular needle in a humongous stack of > needles it is very helpful to move this needle far away from all the > other needles or you can't even see it separately. And again, your response to a yes or no question does not include the word "yes" or "no". >> So are you saying that you *cannot* demonstrate that Gödel proof is >> incorrect by citing a specific error within the proof. It seems to me >> that that's equivalent to saying that Gödel's proof is correct. I'm >> sure that's not what you meant. Did you mean specifically that *you* >> cannot do that? I doubt that that's what you meant either. > > If Gödel's proof is correct except for a single key false assumption > then Gödel's proof is incorrect. And again. >> Are you saying that it's possible for every step of Gödel's proof >> to be valid, but for the proof as a whole to be invalid, yielding a >> false conclusion? If so, that's a remarkable assertion from someone >> who says that a complex system can be complete and consistent. > > A single false premise makes the conclusion unsound. And again. If I ask you a yes or no question, I will ignore any response that does not include the word "yes" or "no", or explain why neither "yes" nor "no" would be meaningful. >> Do you believe there is a specific flaw in Gödel's proof? >> (This question is not about what that flaw is, just whether you >> think there is one.) >> > The definition of incompleteness is its flaw. I'll take that as a yes, but next time I'll ask you to include the word "yes" in your answer if that's what you mean. Really? Is that your whole problem with Gödel's proof, that you don't like the way he defines "incompleteness" (or more likely "Unvollständigkeit")? (Of course the concept existed before Gödel.) > We could define "incomplete" as a term of the art of mathematics such > that every formal system that uses conjunction: "∧", disjunction: "∨", > or negation: "¬" is "defined" to be "incomplete". > > This definition: A theory T is incomplete if and only if there is some > sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ) is equally ridiculous when > all of its implications are very carefully examined. May I presume you have a rigorous definition of "ridiculous"? > Because people are so fully indoctrinated into that definition of > incompleteness it is much much easier to prove that the next level > inference based on that definition: "true and unprovable can coexist" > is impossible. "Incompleteness" is just a word. I understand that you don't like the way it's used. So let's use different words. Perform the following replacements: complete --> blurgicious incomplete --> unblurgicious completeness --> blurgitude incompleteness --> unblurgitude (I've tried to pick words that have no existing baggage, avoiding any preconceived notions about what they mean, so we can define them rigorously and without ambiguity.) Suppose I gave you a copy of (an English translation of) Gödel's proof with the above substitutions performed on it. Adapting a definition from Wikipedia, a set of axioms is blurgicious if and only if, for any statement in the axioms' language, that statement or its negation is provable from the axioms. "Blurgicious" does *not* mean "something is missing". Gödel proved that no sufficiently complex formal system (basically one able to represent the axioms of the natural numbers ) can be both consistent and blurgicious. (I'm assuming you don't have any issues with the word "consistent". If you do, we can invent a replacement for it too.) Now that we're not using the word "complete" or any form of it, how do you refute this version of Gödel's proof? -- Keith Thompson (The_Other_Keith) Keith.S.Thompson+u@gmail.com Working, but not speaking, for Philips Healthcare void Void(void) { Void(); } /* The recursive call of the void */