Importance of this result: Set theory is the axiomatization of mathematics, and without AC no-one seriously doubts its truth, or at least consistency. However, much of mathematics requires AC (eg. every vector space has a basis, every ideal can be extended to a maximal ideal).

You should know the following basic set-theoretic concepts: function, domain, codomain, element, subset, equivalence relation, cartesian product, injection, sur- jection, bijection, image, and preimage (inverse image).

In axiomatic set theory: A set exists if the proposition that asserts its existence is logically true. Moreover within this theory there are only sets so if a formal object is not a set, it does not exist.

106 ZF set theory is formulated in a formal ,rst order language of predicate logic with axioms for equality. 107 !e components of that language L = L P˙ are: 108 (i) set variables;v ,v ,...,v n ,...(for n P )

I formulate the axioms of set theory below. For the moment we have: (AC.) Axiom of Choice (Zermelo) If X is a set of non-empty pairwise disjoint sets, then there is a set Y which has exactly one element in common with each element of X. To complement G¨odel’s theorem, there is also the following result which is beyond this course:

Abstract. In this section we discuss axiomatic systems in mathemat-ics. We explain the notions of “primitive concepts” and “axioms”. We declare as primitive concepts of set theory the words “class”, “set” and “belong to”. These will be the only primitive concepts in our system.

There are two threads that run through basic set theory: foundation via axiomatization, and di erent sizes of in nities. We will see that these two threads are closely connected, and both are very central to set theory. The discovery of di erent sizes of in nities was one of the driving forces behind the development of set theory.