Reddit reviews Set Theory (Studies in Logic: Mathematical Logic and Foundations)

When I took a graduate set theory course, the book used was Kunen's Set Theory (Amazon), which I enjoyed. I've also read through some parts of Jech's Set Theory (Amazon, SpringerLink) and liked what I read.

> I couldn't find any example of that on the web. Do you have a link to such proof?

It's in Kunen's Set Theory chapter 1.

We can prove it ourselves. Assume the axioms of extensionality, pairing, union, and specification. Let A be a set and assume the power set of A exists. Let f : A -> P(A) be any function (formally, we assume it's a set of ordered pairs satisfying the definition of a function with A as its domain and P(A) as its range). Then X = { x in A | x is not in f(x) } is a set by specification. And there is no a in A such that f(a) = X. If there were, this will lead to the usual contradiction (a is in f(a) if and only if a is not in f(a)).

At no point do we appeal to the axiom of foundation. We need the basic axioms to be able to say what it means for f to be a function, and the axiom of specification to be able to conclude that X is a set.

> I meant a universal set U that contains all possible sets within a given set theory. But it's perhaps misleading to call that a universal set.

Again, in any set theory that contains the axioms I mentioned, there is no set of all sets. It wouldn't be a set. In some sense, it's too "big" to be a set. So there is just no such thing.

To piggyback of of /u/blaackholespace, if you're trying to study mathematical logic as a discipline in its own right, I would recommend Enderton's Mathematical Introduction to Logic, which covers the basics of model theory and proof theory up to the Incompleteness Theorems.

If you're looking for a deeper study of set theory (if not, that's cool too), I would recommend Kunen's Set Theory: An Introduction to Independence Proofs.