Martin Hofmann asks: Is it known which functions from N to N are representable in an algebraically closed ccc? (A ccc in which initial algebras for "all" endofunctors exist.) I don't know about algebraically closed ccc's but I should have answered this question in my Durham paper for the algebraically compact case. (Algebraically compact means every endofunctor has both an initial algebra and a final coalgebra and they are canonically isomorphic.) The only algebraically compact ccc is the degenerated category, so we ask for it not to be a ccc but to be a reflective subcategory of an ambient ccc. To get off the ground for a (flat) natural numbers object, N, we ask that the algebraically compact category have finite coproducts. Then the answer is, essentially, that every recursive function appears as an endomorphism of N. To make this precise, define a "point" of N to be a map thereto from the terminator, a "standard point" to be either the 0-point or one of its successors. Any endomorphism on N induces a partial endomorphism on the standard points. The result is that every recursive partial function is induced by an endomorphism on N. Since there is something of a free structure for this theory we can not expect to get more than the recursive. To return to the original question: does someone have a counterexample to back up Hans Dybkjaer's comment: Though I do not remember any reference, I do not think that more expressive power in terms of functions N->N is gained by adding other initial algebras =========================================================================