On Sat, 31 Jan 2009, Vaughan Pratt wrote:
I'm not aware of any reason why a topos with a Cantor set object K has to also have a natural number object N, though I'm not enough of a topos hacker myself to know how to produce one with K but without N (but would be happy to learn). Does such a topos exist in nature? And what can be said of the free topos with Cantor set object?
A topos with a Cantor set object (i.e. a final coalgebra for FX = X+X) necessarily has a natural number object. Observe that the Cantor set K necessarily has a point (since 1 has an F-coalgebra structure), so the isomorphism K+K \cong K yields a monomorphism K \to K and a point disjoint from its image. From there on, use Corollary D5.1.3 in the Elephant. Peter Johnstone