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 (Membership algorithm) Date: Thu, 16 Jul 2020 21:00:28 -0700 Organization: None to speak of Lines: 48 Message-ID: <87zh7yyen7.fsf@nosuchdomain.example.com> References:

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<87y2nkguqv.fsf@bsb.me.uk> <87h7u7h54e.fsf@bsb.me.uk> <8rKdnZAbAoVsOo3CnZ2dnUU7-efNnZ2d@giganews.com> <87d04vyzd2.fsf@nosuchdomain.example.com> Mime-Version: 1.0 Content-Type: text/plain Injection-Info: reader02.eternal-september.org; posting-host="22913342158febd17ce42e5a39ff8077"; logging-data="10736"; mail-complaints-to="abuse@eternal-september.org"; posting-account="U2FsdGVkX1+1qnlGrS+dnJtuv7fWeAc2" User-Agent: Gnus/5.13 (Gnus v5.13) Emacs/26.3 (gnu/linux) Cancel-Lock: sha1:jPBUC3gi/vMZQUllMyLeF6AsbQY= sha1:/53OTWP/AE7RquoV+i5N9TJkhvU= Xref: csiph.com comp.theory:21729 comp.ai.philosophy:22075 comp.ai.nat-lang:2450 olcott writes: > On 7/16/2020 3:32 PM, Keith Thompson wrote: >> olcott writes: >> [...] >>> I understand that Q defines natural numbers so that Q does not assume >>> natural numbers. Q defines them as whatevers that have a successor >>> function. >> >> Are you sure about that? Are you sure that there's no possible model >> that satisfies the axioms of Q but is incompatible with the properties >> of the natural numbers? >> >> Let's consider a collection of axioms simpler than those of Q: >> >> - A set contains a member "0". >> - N + 0 = N. >> - If N is a member of the set, then N+1 is a member of the set. >> - Addition is defined as ... >> >> Have I defined the natural numbers? No. The natural numbers >> satisfy all those axioms, but so do the integers, and so do the >> reals, > > Give me the Real number successor to 1.5. 2.5 I didn't use the word "successor" in my axioms. I did define "N+1" (you can call that "successor" if you like), and though I didn't say so explicitly, it's just addition. Perhaps you were thinking that 1.5 has no successor, since for any real number X > 1.5, there's another real number Y with Y > 1.5 and Y < X. But I never said that N+1 is the *immediate* successor of N, and I didn't define a ">" relation. >> complex numbers, and quaternions. >> >> Are there sets that satisfy Q while violating some of the properties >> of the natural numbers? I don't know. Do you? If so, how do >> you know? >> >> (I'm probably being imprecise about sets, models, and so on.) -- 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 */