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.
While we're about it, let's include "greater than 1" in the definition of regular cardinal please, so that 2 and omega are regular but 0 and 1 are not. This would ensure that, for each regular cardinal kappa, the endofunctor on Set taking A to the set of subsets (or nonempty subsets) of A of size < kappa is a monad. To put it another way, regularity means in essence "closed under dependent sum", and singleton is the unit of dependent sum. Paul 16-Mar-2005 11:16:08 -0400,5626;000000000000-00000000