1. there are only countably many objects x in M such that M(x) "∈" M(2^N).
2. cardinality inside the model no longer has any meaning once you go out of the model, since it's just another first-order logic formula.
3. let SM={ M(x) "∈" M(N) : M(x) "∉" f(M(x))}
4. SM doesn't need to correspond to anything inside M, since SM is a set defined outside M.