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, 10 Jul 2020 15:06:15 -0700 Organization: None to speak of Lines: 108 Message-ID: <878sfr6n54.fsf@nosuchdomain.example.com> References: <87k0zc8ps5.fsf@nosuchdomain.example.com> <87v9iv6t9z.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="afce2cd57a04b7ce2f64abcfcc7ff62f"; logging-data="12044"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1/YNtVS2NzYzQMm0c7uu70x" User-Agent: Gnus/5.13 (Gnus v5.13) Emacs/26.3 (gnu/linux) Cancel-Lock: sha1:OJCK6v0cdo+zVYm0OHVfJt2Ikcw= sha1:WstoBlvnA/El4uF8XZx+ccE5rDw= Xref: csiph.com comp.theory:21568 comp.ai.philosophy:21887 comp.ai.nat-lang:2320 olcott writes: > On 7/10/2020 2:53 PM, Keith Thompson wrote: >> olcott writes: >>> On 7/9/2020 2:14 PM, Keith Thompson wrote: >>>> olcott writes: >>>>> On 7/9/2020 8:40 AM, André G. Isaak wrote: >>>> [...] >>>>>> I've asked you repeatedly about Robinson's arithmetic, in which x + >>>>>> y = y + x is not provable. Neither is ¬(x + y = y + x) provable. The >>>>>> law of the excluded middle demands that one of those be true, so >>>>>> there exists a true statement in Q which is not provable in Q. >>>>>> >>>>>> And one can prove that x + y = y + x is true in Q. You just can't >>>>>> prove it from within Q. >>>>> >>>>> That is the exactly same key mistake that you, Tarski and presumably >>>>> Gödel made. How do we know that it is true IN Q when it is not >>>>> provable IN Q (We look outside of Q). THEN IT IS NOT TRUE IN Q, IT IS >>>>> ONLY TRUE OUTSIDE OF Q. >>>> >>>> If it is not true in Q, then there are values x and y in Q such that >>>> x + y = y + x is false in Q. >>>> >>>> In fact there are no such values. (You could refute that if you could >>>> provide such values.) >>>> >>>> I'm assuming that "x + y = y + x is true in Q" and "x + y = y + x is >>>> false in Q" are the only possibilities (law of the excluded middle). >>>> Do you accept that assumption? > > No I do not accept that assumption. Q does not know about the > commutative property of addition so it is neither true nor false in Q. > >>> >>> This is my current best guess of the correct use of the term >>> satisfiable if the term satisfiable can even be applied to a single >>> theory: >>> >>> ∃φ (Q ⊢ "x + y = y + x") would seem to be unsatisfiable in Q. >>> ∃φ ¬(Q ⊢ "x + y = y + x") would also seem to be unsatisfiable in Q. >>> >>> This would seem to indicate that Q is incomplete relative to commutativity. >>> >>> I am certain that the ideas are correct. I am uncertain if my use of >>> the term unsatisfiable corresponds to its conventional use. >>> >>> I am certain that my use of the term incomplete correctly augments the >>> conventional use of the term such that my use is more correct than the >>> conventional use. >> >> And this is an example of why trying to have a conversation with you is >> so frustrating. >> >> I asked what I thought was a straightforward yes or no question, > > I answered with all of the reasoning behind the correct answer. > It is like you asked me are their any five million pound giant humans? > I answer that there is no animal that weighs more than 330,000 lbs. > Then you said I did not answer your question. That kind of answer is not useful if the person asking the question does not accept understand the reasoning behind the answer. In your example, for that to be an answer I'd have to accept that humans are animals and that five million pounds is more than 333,000 pounds. Of course I do accept both of those, but that's not the case with your statements about mathematical logic, and you should stop assuming that it is. I won't address (at least not in this post) whether I don't understand your reasoning because I'm too stupid to understand your brilliance or because you're wrong. You seem to be frustrated that the rest of us either don't understand or don't accept your claims. I get that, and I believe you're sincere. But if your intent is to communicate, you must at least *accept the observed fact* that the rest of us either don't understand or don't accept your claims. If I ask you a yes or no question (which I've spent time carefully constructing for the purpose of eliciting information), it would be helpful if your answer would include the word "yes" or "no", preferably at the beginning -- or an assertion, or ideally an explanation, that neither "yes" nor "no" would be meaningful. Whatever else you want to say after that is fine. For example: Q: Are their any five million pound giant humans? A: No, because there is no animal that weighs more than 330,000 lbs. Q: OK, but how does your answer follow from your following statement? What are you assuming? ... and so on. >> "Do you >> accept that assumption?". Your response did not include the word "yes" >> or "no", nor did it attempt to demonstrate that neither "yes" nor "no" >> would be a meaningful answer. Instead you wrote several paragraphs >> about the meaning of "satisfiable". >> >> By all means, write all you like about the meaning of "satisifiable", >> but please don't do so in a context that makes it look like you're >> trying to answer my question. Perhaps what you wrote has some relevance >> to what I asked, but I don't see it. >> >> You have not answered my question. "Yes" or "No" would be an answer. -- 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 */