Path: csiph.com!aioe.org!.POSTED.u0Pso9KBOjSJs8bnrzkLzw.user.gioia.aioe.org!not-for-mail From: Louis Valence Newsgroups: comp.lang.c++,comp.ai.philosophy,comp.lang.prolog,comp.lang.lisp Subject: Re: Provably unprovable eliminates incompleteness Followup-To: comp.lang.lisp Date: Mon, 09 Sep 2019 17:43:24 -0300 Organization: Aioe.org NNTP Server Lines: 55 Message-ID: <86v9u1gpzn.fsf@example.com> References: <581781046.589713241.741596.DerTopper-web.de@news.eternal-september.org> NNTP-Posting-Host: u0Pso9KBOjSJs8bnrzkLzw.user.gioia.aioe.org Mime-Version: 1.0 Content-Type: text/plain X-Complaints-To: abuse@aioe.org User-Agent: Gnus/5.13 (Gnus v5.13) Emacs/24.3 (windows-nt) Cancel-Lock: sha1:1bNg9g5Um4ADwPBPkvI2dYazNIs= X-Notice: Filtered by postfilter v. 0.9.2 Xref: csiph.com comp.lang.c++:68289 comp.ai.philosophy:19092 comp.lang.prolog:9616 comp.lang.lisp:33258 Stuart Redmann writes: > Dhu on Gate wrote: [...] >> Since this is unreadable by peons, I can only speculate, >> but provably unprovable sounds like a useless tautology. >> >> Dhu > > It isn't. It rather shows that - like the speed of light in physics - there > are some rather unexpected limitations to what is possible in mathematics. > For many years mathematicians believed that, given a set of axioms, every > possible theorem in the given field can be proven to be either true or > false based on logical deduction from these axioms. > > Goedel showed that this is not always the case (see the continuum > hypothesis). This implies that there are things in mathematics that cannot > be proven. This culminated in Goedels incompleteness theorem, which is > considered by many mathematicians as a severe throwback in maths. Despite my very limited understanding of logic, mathematics and everything else, I can't agree with you. There are some limitations to what is possible in formal logic; the same cannot today be said about mathematics. There are things in some formal logic systems that cannot be proven, but AFAIK there is no known result to the effect of what mathematics can or cannot prove. Peter Smith in ``An Introduction to Godel's Theorems (ISBN: 978-0-511-35096-2)'', on section 1.2, page, writes: --8<---------------cut here---------------start------------->8--- Suppose we try to specify a suitable axiomatic theory T that seems to capture the structure of the natural number sequence and pin down addition and multiplication (and maybe a lot more besides). Then Godel gives us a recipe for coming up with a corresponding sentence G(T), couched in the language of basic arithmetic, such that (i) we can show (on very modest assumptions, e.g. that T is consistent) that neither G(T) nor ~G(T) can be derived in T, and yet (ii) we can also recognize that, at least if T is consistent, G(T) will be true. --8<---------------cut here---------------end--------------->8--- So, Godel's recipe itself shows T is incomplete, meaning T cannot prove (as true or false) everything that can be stated in T. One could think that, therefore, we don't know whether the statement G(T) is true or false. Peter Smith adds that we can also recognize that, assuming T is consistent, then G(T) will be true. This /recognition/ is not backed up by T because T only asserts what it can --- and T cannot assert G(T). We may say we have a precise definition of a system of logic is, but we cannot say we have a precise definition of what mathematics is. What we can say is that if mathematics can be perfectly equated to logic, then we don't have any idea of what this system of logic would be to be equated to what we think mathematics is.