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, 09 Jul 2020 12:14:02 -0700 Organization: None to speak of Lines: 41 Message-ID: <87k0zc8ps5.fsf@nosuchdomain.example.com> References: Mime-Version: 1.0 Content-Type: text/plain; charset=utf-8 Content-Transfer-Encoding: 8bit Injection-Info: reader02.eternal-september.org; posting-host="96132afdf8c69f443faa5b2708058e93"; logging-data="19742"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX18QTdutziOi8Sl+GnXXtHzT" User-Agent: Gnus/5.13 (Gnus v5.13) Emacs/26.3 (gnu/linux) Cancel-Lock: sha1:ThnQjUTFii4ncXhTWDO6lP+o8SY= sha1:0J0YGeUuwDjK0fbFlH7Iz8i0ld8= Xref: csiph.com comp.theory:21532 comp.ai.philosophy:21816 comp.ai.nat-lang:2286 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? Please indicate whether you agree or disagree with each of the following statements: 1. x + y = y + x is true in Q 2. x + y = y + x is not true in Q 3. x + y = y + x is false in Q 4. x + y = y + x is provable in Q I claim that 1 is true, 2 is false, 3 is equivalent to 2 and therefore also false, and 4 is false. What say you? -- 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 */