Hi,

I know it is possible to construct a model with a non trivial ultrafilter u and I want to show u is such that for all fonction f from $\mathbb{N}$ to $\mathbb{N}$ there is an element of u such that f is constant or strictly monotonic on u (we will say that u is a selective ultrafilter). I know that we must go from a model of secondary order logic M with comprehension schema and dependant choice axiom and with set of individuals $\mathscr{P}(N)$ and consider N=M[G] where G is a generic which is a non trivial ultrafilter in N on $\mathbb{N}$ (G exists) but I don't know how to show u is selective.

Thanks for your help and suggestions.