From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/280 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: RE: question on finiteness in toposes Date: Wed, 15 Jan 1997 10:33:21 -0400 (AST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241016867 25013 80.91.229.2 (29 Apr 2009 14:54:27 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:54:27 +0000 (UTC) To: categories Original-X-From: cat-dist Wed Jan 15 10:33:42 1997 Original-Received: by mailserv.mta.ca; id AA23534; Wed, 15 Jan 1997 10:33:21 -0400 Original-Lines: 30 Xref: news.gmane.org gmane.science.mathematics.categories:280 Archived-At: Date: Wed, 15 Jan 97 10:19 GMT From: Dr. P.T. Johnstone Not an answer to Bill's question (which I agree is an important one), but a minor correction. Bill wrote: While the K/S definition is right for the construction of the object classifier over an arbitrary base topos (as Gavin showed) and hence for classifiers for various kinds of finitary algebras over an arbitrary base topos, It isn't, and he didn't. Gavin used finite cardinals to construct the object classifier over an arbitrary base topos with NNO (and I subsequently extended the construction to finitary algebraic theories), but it doesn't work over a topos without NNO (and in particular it can't be made to work using K-finiteness). Andreas Blass showed that the existence of an object classifier for toposes over E implies that E has a NNO. Incidentally, I think it is correct to give credit to Kuratowski for the notion of K-finiteness. It's true that Sierpinski's paper was earlier, but his definition was a "global" one (i.e. he defined the class of all finite sets as the sub-semilattice of the universe generated by he singletons), whereas Kuratowski made the crucial observation that the finiteness of a particular set X can be determined locally (i.e. within the power-set of X), without which the notion could never have been imported into topos theory. Peter Johnstone