1) Prove that there is no function f with domain ω such that f (n+) ∈ f (n) for all n ∈ ω. [Hint: apply the Axiom of Foundation to ran f .] Deduce that, for any set x, it is false that x ∈ x.
2) Prove, using the Principle of Induction and the fact that each n ∈ ω is a transitive set, that n ∈ n is false for every natural number n.
Presumably, the idea in the first question is to show that if it were the case that there were such a function, then the range of f would contradict the axiom of Foundation, but I'm not entirely sure how to do the details.
Any help would be greatly appreciated. Many thanks.