From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3419 Path: news.gmane.org!not-for-mail From: "Mamuka Jibladze" Newsgroups: gmane.science.mathematics.categories Subject: Morley derivative as non-classical Cantor-Bendixson derivative? Date: Mon, 4 Sep 2006 11:53:05 +0400 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;format=flowed; charset="iso-8859-1"; reply-type=response Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019292 8653 80.91.229.2 (29 Apr 2009 15:34:52 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:34:52 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Mon Sep 4 09:32:37 2006 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 04 Sep 2006 09:32:37 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1GKDb4-0000L7-PW for categories-list@mta.ca; Mon, 04 Sep 2006 09:30:39 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 9 Original-Lines: 87 Xref: news.gmane.org gmane.science.mathematics.categories:3419 Archived-At: Mention of the Morley derivative by Vaughan Pratt reminded me of my recent suspicion that I think would be natural to share here. Is there a chance to obtain the Morley derivative by performing construction of the Cantor-Bendixson derivative internally on an appropriately chosen internal locale in the topos of set-valued functors on an appropriate subcategory of spaces and continuous maps? Mamuka Jibladze ----- Original Message ----- From: "Vaughan Pratt" To: Sent: Sunday, September 03, 2006 10:59 PM Subject: categories: Re: no membership-respecting morphisms > My request for precursors to Mike Barr's deprecation of set membership > seems to have set loose a thread that has veered off from set theory > into model theory. The following extract from the introduction to > Gerald Sacks' "Saturated Model Theory" (335 pages, W.A. Benjamin, 1972) > serendipitously ties up this loose thread while at the same time > promising that category theory offers deeper insight into categoricity, > a central notion of model theory, than the alternatives. > > "It is true that model theory bears a disheartening resemblance to set > theory, a fascinating branch of mathematics with little to say about > fundamental logical questions, and in particular to the arithmetic of > cardinals and ordinals. But the resemblance is more of manners than of > ideas, because the central notions of model theory are absolute, and > absoluteness, unlike cardinality, is a logical concept. That is why > model theory does not founder on that rock of undecidability, the > generalized continuum hypothesis, and why the Los conjecture is > decidable: A theory T is k-categorical if all models of T of cardinality > k are isomorphic. Los conjectured and Morley proved (Theorem 37.4) that > if a countable theory is k-categorical for some uncountable k, then it > is k-categorical for every uncountable k. The property 'T is > k-categorical for every uncountable k' is of course an absolute property > of T. > > The notion of rank of 1-types was invented by Morley to prove Los's > conjecture. There are proofs of it that make no mention of rank, but > they leave one ill-prepared to prove Shelah's uniqueness theorem > (Section 36). I have made rank a central idea of the book, because it > is the central idea of current model theory. ... Morley's notion of > rank was inspired by the Bendixson differentiation of a closed subset of > a compact Hausdorff space; however, the Morley derivative differs from > the Cantor-Bendixson derivative in that the former commutes with the > inverse limit operation. The Morley derivative is expounded in section > 29 as a transformation which acts on functors of a class common in model > theory. One advantage of a category theoretic treatment of Morley rank > is that it applies equally well to other notions [Shelah] of rank of > 1-types. Section 25 reviews the apparatus of category theory needed in > section 29." > > The difference between this recommendation of category theory for model > theory and (for example) the literature on accessible categories is that > Sacks was not a card-carrying category theorist but a recursion > theorist. While category theory has no bias towards Goedel's notion of > absoluteness (that I'm aware of), it seems reasonable to infer from > Sacks' acceptance of CT that neither is CT biased away from absoluteness > but rather is a neutral general-purpose tool. > > Vaughan Pratt > > > V. Schmitt wrote: >> Dear Paul, as far as i remember model theorists have >> an extremely elegant way of comparing structures: >> no morphisms there but "games" of finite isomorphism >> extensions (see Fraisse' and Ehrenfeucht for the game >> aspect or B.Poizat's book) >> All the first order syntax is subsumed by those games. >> I quite like categories and for sure I do not know everything, >> but I have not seen so far a convincing categorical counterpart >> for these games. >> (And model theory is good stuff!) >> >> Best, >> Vincent. > > >