0. Show that there exist infinitely many finite sets that are not transitive and infinitely man infinite sets that are not transitive.
1. Show: A set X is transitive if and only if X is a subset of P(X).
2. Show: if every X *belongs to* S is transitive, then union(S) is transitive.
3. Show: an ordinal α is a natural number if and only if every nonempty subset of α has a greatest element.
4.Show: If a set of ordinals X does not have a greatest element, then sup X is a limit ordinal.
5. Show: If X is a nonempty set of ordinals, then intersection(X) is an ordinal. Moreover, intersection(X) is the least element of X.
Unfortunately, your question requires a lot of work and cannot be done for free. Submit it with all requirements as an assignment to our control panel and we'll assist you.