A news.group post I snagged to think about later … related to the question of "proving negatives."

Xref: sn-us news.groups:543329 Path: sn-us!sn-feed-sjc-01!sn-us!sn-feed-sjc-04!sn-xt-sjc-10!sn-xt-sjc-01!sn-xt-sjc-13!supernews.com!news.glorb.com!postnews.google.com!19g2000hsx.googlegroups.com!not-for-mail From: aatu.koskensilta@xortec.fi Newsgroups: news.groups Subject: Re: Policy Discussion -> Re: Brainstorming Session - "Obvious Groups" Date: Tue, 02 Oct 2007 06:35:22 -0700 Organization: http://groups.google.com Lines: 36 Message-ID: <1191332122.544516.59560@19g2000hsx.googlegroups.com> References: <tskirvin.20070911173402$048f@cairo.ks.uiuc.edu> <871wcx3914.fsf@huxley.huxley.fi> <46ee5e17$0$47135$892e7fe2@authen.yellow.readfreenews.net> <87k5qp1rmy.fsf@huxley.huxley.fi> <46ee62ad$0$47143$892e7fe2@authen.yellow.readfreenews.net> <87ejgx1qhg.fsf@huxley.huxley.fi> <46fbc74b$0$32559$4c368faf@roadrunner.com> <1190986310.390055.37540@o80g2000hse.googlegroups.com> <46ffafbe$0$7464$4c368faf@roadrunner.com> <87r6kgnsah.fsf@huxley.huxley.fi> <47015ac2$0$32478$4c368faf@roadrunner.com> NNTP-Posting-Host: 81.197.123.102 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" X-Trace: posting.google.com 1191332122 30382 127.0.0.1 (2 Oct 2007 13:35:22 GMT) X-Complaints-To: groups-abuse@google.com NNTP-Posting-Date: Tue, 2 Oct 2007 13:35:22 +0000 (UTC) In-Reply-To: <47015ac2$0$32478$4c368faf@roadrunner.com> User-Agent: G2/1.0 X-HTTP-UserAgent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.7.9) Gecko/20050711 Firefox/1.0.5,gzip(gfe),gzip(gfe) Complaints-To: groups-abuse@google.com Injection-Info: 19g2000hsx.googlegroups.com; posting-host=81.197.123.102; posting-account=ps2QrAMAAAA6_jCuRt2JEIpn5Otqf_w0 dvus wrote: > Aatu Koskensilta wrote > > > Mathematical logicians are paid by other mathematical logicians whose > > work is equally uninteresting to any sane person. Perhaps you would > > like to know why, on the assumption that every set is constructible, > > there is no finitely axiomatisable higher-order theory that is > > complete but not categorical? > > Quantum erasers? Alas, no. The proof is rather simple. First recall that there is a definable well-ordering < of the constructible universe. Observe then that for finitely axiomatisable higher-order theories T the property of a model being isomorphic to <-least model of T is expressible by a purely logical sentence. Let now A be this sentence. Since the <-least model M of T is, trivially, a model of T, A holds in M. So T can't refute A, which means A is actually provable since T is complete, and thus all models of T are isomorphic. This concludes the sketch of the proof that, assuming V=L, all complete finitely axiomatisable higher-order theories are categorical. This is the sort of exciting stuff we logicians like to prattle about. In set theory we often also talk about the universal weasel, premice, the zero-handgranade, the axiom of playful universe, ineffable cardinals, small large cardinals, large large cardinals, indestructibility, forcing, games between God and Devil, etc. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus