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.
I see what's happened- the symbols are displayed on my screen, but clearly not on everyone's... I thought I couldn't see what the problem was...
In normal language, the questions are:
1) Prove that there is no function f with domain ω (omega) such that f (n+) is a member of f (n) for all n in ω (omega). Deduce that, for any set x, it is false that x is a member of itself. (where n+ is the sucessor of n)
2) Prove, using the Principle of Induction and the fact that each n in ω (omega) is a transitive set, that the statement "n is a member of itself" is false for every natural number n. (i.e. prove without using the Axiom of Foundation)
Sorry for the confusion.