From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2308 Path: news.gmane.org!not-for-mail From: "Elwood Wilkins" Newsgroups: gmane.science.mathematics.categories Subject: Re: Topos logic arises naturally Date: Fri, 30 May 2003 12:35:06 +0100 Message-ID: <002801c326a1$05dc0000$f112fea9@essex.ac.uk> References: <3ED34194.9050005@informatik.uni-bremen.de> Reply-To: "Elwood Wilkins" NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241018564 3475 80.91.229.2 (29 Apr 2009 15:22:44 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:22:44 +0000 (UTC) To: "Categories" Original-X-From: rrosebru@mta.ca Fri May 30 23:18:07 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 30 May 2003 23:18:07 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19Lvrf-0002EB-00 for categories-list@mta.ca; Fri, 30 May 2003 23:12:59 -0300 X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.00.2615.200 X-MimeOLE: Produced By Microsoft MimeOLE V5.00.2615.200 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 58 Original-Lines: 65 Xref: news.gmane.org gmane.science.mathematics.categories:2308 Archived-At: Isabelle's internal logic is not constructive. The existential clause of the BHK-interpretation is violated so that a consquence of "all unicorns have horns" is that "some unicorn has a horn". The moral (biased towards theorists): software engineering considerations are not enough, a coherant philosophy of mathematics is also needed. Elwood ----- Original Message ----- From: Till Mossakowski To: Categories Sent: Tuesday, May 27, 2003 11:44 AM Subject: categories: Topos logic arises naturally > You see: even with just aesthetic and software engineering > considerations you are eventually lead to topos logic... > > -- > Till Mossakowski Phone +49-421-218-4683 > Dept. of Computer Science Fax +49-421-218-3054 > University of Bremen till@tzi.de > P.O.Box 330440, D-28334 Bremen http://www.tzi.de/~till > > Message-Id: <200305270616.h4R6G3w8008134@plxc2089.pdx.intel.com> > To: Randy Pollack > cc: John Harrison , isabelle-users@cl.cam.ac.uk > Subject: Re: HOL without description operators > Date: Mon, 26 May 2003 23:16:03 -0700 > From: John R Harrison > Sender: Larry Paulson > > Hi Randy, > > | Perhaps more usefully, how do you suggest I do this task of developing HOL > | without description operators. > > You might find it interesting to look at HOL Light. This gives a > different axiomatization of the HOL logic, developed precisely because I > was dissatisfied with the one in the original HOL system, on which I > believe the Isabelle/HOL logic is based. > > Although I do eventually introduce the description operator, quite a lot > of the basic logic --- including the automation of inductive definitions > --- is developed without it (and indeed without excluded middle or > extensionality). You may find it a more congenial starting-point. > > As Konrad Slind later pointed out, what I settled on is remarkably close > to the internal logic of a topos, as presented for example in Lambek and > Scott's book. This was a surprise to me since at that time I knew next to > nothing about toposes and was motivated by a mixture of aesthetic and > software engineering considerations. > > Cheers, > > John. > > > >