In fact, the mere existence of a subset of N so that you can always insist on the appearance of some element from that subset is already provably equivalent to the 1-consistency of Mahlo cardinals of finite order. In the definition of apporoximate fixed points, we can use other functions instead of or in addition to summation. We can use finitely many functions of several variables on N that are strictly increasing in each argument. More generally, we can use finitely many functions of several variables that are increasing in each argument and dominating. This amounts to a higher order Lipschitz condition on phi.
C be three sets.
We can strengthen Proposition 2 considerably in many ways. Here is an example.