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: Fri, 17 Jul 2020 01:04:24 -0700 Organization: None to speak of Lines: 262 Message-ID: <87ft9qy3cn.fsf@nosuchdomain.example.com> References: <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> <87sgdrz49w.fsf@nosuchdomain.example.com> <874kq7yug9.fsf@nosuchdomain.example.com> <4dKdnXavpI9eu4zCnZ2dnUU7-S3NnZ2d@giganews.com> Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: 8bit Injection-Info: reader02.eternal-september.org; posting-host="22913342158febd17ce42e5a39ff8077"; logging-data="11292"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX18Mc+4gVGMDACvWBfMtgOnd" User-Agent: Gnus/5.13 (Gnus v5.13) Emacs/26.3 (gnu/linux) Cancel-Lock: sha1:Jv8JjBwrlv0lt2KxxHsloBIrEV0= sha1:395+yOqcQzh4zt1Fw9z8rYXZUvo= Xref: csiph.com comp.theory:21738 comp.ai.philosophy:22083 comp.ai.nat-lang:2458 olcott writes: > On 7/16/2020 5:19 PM, Keith Thompson wrote: >> olcott writes: >>> On 7/16/2020 1:46 PM, Keith Thompson wrote: >>>> 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 that is even better. I thought that he only concluded that some >>> formulas are neither provable nor disprovable. >> >> Quoting https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems : >> >> The first incompleteness theorem states that no consistent system >> of axioms whose theorems can be listed by an effective procedure >> (i.e., an algorithm) is capable of proving all truths about the >> arithmetic of natural numbers. For any such consistent formal >> system, there will always be statements about natural numbers >> that are true, but that are unprovable within the system. The >> second incompleteness theorem, an extension of the first, >> shows that the system cannot demonstrate its own consistency. >> > > Show me where Gödel say that in his actual paper: > https://mavdisk.mnsu.edu/pj2943kt/Fall%202015/Promotion%20Application/Previous%20Years%20Article%2022%20Materials/godel-1931.pdf I've already said that I haven't read Gödel's proof and probably wouldn't understand it. (Possibly I could if I devoted the time and effort to it, but I'm not planning to.) You're the one who's making the grandiose claims. Is the statement in Wikipedia wrong? >> Your "OK that is even better" remark seems to imply that you didn't know >> that, which is frankly staggering. You've spent years claiming to have >> refuted Gödel's proof, but you don't even know what he proved. > >>>> 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". >>> >>> How can you show that Gödel really showed that true and unprovable >>> formulas exist. In other words and this is not merely an >>> interpretation that someone added later on. >> >> By reading and understanding his proof. I'm honestly not prepared >> to do that myself. I think that some other participants here have, >> and I'll let them comment further if they choose to do so. >> >> Here's another yes or no question: Have you read Gödel's proof? > I just posted a copy of his whole paper. Not an answer. You didn't post a copy. You posted a link. So did I, by quoting your post. I haven't read Gödel's proof. Have you? > I already proved that he himself did not understand it because he > directly contradicted himself in his footnote 15. No, he didn't. >>> The best way would be to quote his paper where he would say: >>> "I just proved that true and unprovable formulas exist". >>> >>> Even his use of natural language seems to be about as convoluted as he >>> can possibly make it. If he were to say: "I just proved that true and >>> unprovable formulas exist" it would take him at least fifteen pages. >>> >>>>>>>> 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"? >>> >>> Yes >> >> Well, I wasn't expecting that answer. >> >> You seem to be saying that it would *not* be possible to refute Gödel's >> proof by finding a specific flaw in it. Is that really what you mean? > > Based on his incorrect foundation his conclusion would probably be > correct. I have been able to reconstruct enormously simpler examples > of self contradictory expressions that "prove" that their formal > system is "incomplete". What "incorrect foundation" are you talking about? Is anything more than simply *assuming* that all true statements are provable, and therefore Gödel must obviously be wrong? > He admitted in his paper that any self-contradictory expression can be > used to derive the same proof: > > 14 Every epistemological antinomy can likewise be used > for a similar undecidability proof > >>>>> "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. >>> >>> I am fully refuting your general whole point as it can be applied to >>> Gödel or anything else. >>> >>> If it can be proved that a conclusion is incorrect then there must >>> have been some error somewhere in the reasoning that lead to the >>> conclusion. No need to even look at this reasoning as long as its >>> conclusion can be proved to be incorrect. >> >> You have not answered my question. >> >> 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? Yes or no, please. > > Not quite. Every single step besides a single false premise. What false premise? >>>>>> 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.) >>> >>> If he proved that there are true and unprovable formulas once you >>> understand how True(x) really works you will see that it is the same >>> as if he proved 3 > 7, utterly impossibly correct. >> >> Yes or no, please. You know, it's possible to trim quoted material when you post a followup. It's even considered polite to do so if you're not going to say anything about it. I asked you a question. You are of course under no obligation to answer it, but if you're not going to answer it please stop quoting it. Here's the question again: Is that your whole problem with Gödel's proof, that you don't like the way he defines "incompleteness" (or more likely "Unvollständigkeit")? >>>>> 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"? >>> >>> Ideas deserving of disparagement? >> >> I'll take that as a no, you don't have a rigorous definition of >> "ridiculous". >> >> In my opinion, concepts like "ridiculous", "deserving", and >> "disparagement" have no legitimate place in this discussion. Again, you quote large blocks of text and ignore them. > All that anyone had to to do prevent this error is follow the sound > deductive inference model. > > It seems a little too weird that none of the greatest mathematicions > in the world could apply junior high school logic to the definition of > incompleteness. Yeah, that seems weird, doesn't it. It almost seems as if Gödel was actually right. A definition merely tells us how a word is being used. Is the concept to which the word "undecideability" is applied a valid concept? (Note that a valid concept can be something that doesn't necessarily exist; we have words for "centaur" and "unicorn" and we know what they mean.) Are you *really* basing all this on your dislike of the way a word was used? Does your argument hold together if we use a different word for that same concept? > When you start with truth and only apply truth preserving operations > then you necessarily always must end up with truth. If there is any > break in the inference chain then you simply do not end up with truth. -- 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 */