Vaughan Pratt wrote in part:
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.
Without using any form of Choice (not even Excluded Middle), you can define X+ (for any set X) as the set of ordinal numbers (where an ordinal number is a set equipped with a well founded, transitive, extensive binary relation; modulo isomorphism) that can be injected (as sets) into X; this is a quotient set (as we mod out by isomorphism of sets equipped with binary relations) of a subset (as we pick only the well founded, transitive, extensive binary relations) of the power set of X + X^2 (as we pick a subset of X and a binary relation on X that we restrict to the chosen subset). There is a big analogy that runs like this: a+ : 2^a :: aleph : beth :: Burali-Forti : Cantor :: more? X+ is the _Hartogs_number_ of X. -- Toby 16-Mar-2005 11:18:09 -0400,5075;000000000001-00000000