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blog:aatu [2017/07/13 20:33] (current)
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 +A news.group post I snagged to think about later ... related to the question of [[http://​www3.canisius.edu/​~moleski/​proof/​provenegs.htm|"​proving negatives."​]]
 +
 +<​code>​
 +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
 +
 +</​code>​
 +
  
 
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