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