An infinite cardinal a is called a strong limit cardinal, if b<a implies 2^a<b. For every strong limit cardinal a the category of all sets of cardinality < a is en elementary topos,
In the definition of strong limit cardinal (with 2^b < a of course), some authors set a lower bound of either nonzero or uncountable on strong limit cardinals, seemingly arbitrarily. In view of the above connection, presumably a toposopher would argue for the former lower bound and against the latter. (One certainly wouldn't want to overlook the topos of finite sets, and, lacking Omega, the empty category cannot be a topos.) The inclusion of "nonzero" somewhere in the definition of strong limit cardinal would therefore nail down that detail at least for topos theory. That goes for Elephant A.2.1.2, which overlooks the case \kappa=0 by giving the above as the definition of what Peter J. calls a "limit power cardinal." (This seems an odd name for "strong limit cardinal" btw, given that no strong limit cardinal can be a power cardinal in the sense of arising as 2^a for some cardinal a. "Power limit cardinal" maybe, but why proliferate terminology?) If one were looking for differences in outlook between set theory and topos theory, the stranger notion of limit cardinal might be a more fruitful object to contemplate. A limit cardinal is defined as for a strong limit cardinal with a+ in place of 2^a, where a+ denotes the least cardinal strictly greater than a. Whereas 2^a is a perfectly good notion in a topos, with Cantor's theorem making it a fine successor operation for transfinite cardinals, a+ is the sort of thing I would have thought only a fan of ZFC could love. Or is there in fact a constructively acceptable notion of limit cardinal that is weaker than strong limit cardinal? For a constructivist the very need for "strong" in the definition of limit cardinal seems like a bad hangover from set theory. On the relevance of replacement to the relationship between set theory and category theory, how much better off are "working mathematicians" with replacement as a tool of their trade than with category theory? My impression is that category theory is used increasingly more in applied mathematics these days. Is there a comparable trend for replacement? Vaughan Pratt 14-Mar-2005 20:16:35 -0400,1428;000000000001-00000000