Students in the course should familarise themselves with certain concepts in advance. The time of the course is not sufficient to cover all details of the following material.

The ten axioms of ZFC are listed below. (Strictly speaking, the ZFC axioms are just strings of logical symbols. What follows is only an attempt to express the intended meaning of these axioms in English.) Note that Separation and Replacement are both axiom schemata. Each axiom has its own article giving additional information.

1.Axiom of extensionality: Two sets are the same if and only if they have the same elements.

2.Axiom of empty set: There is a set with no elements. We will use {} to denote this empty set.

3.Axiom of pairing: If x, y are sets, then so is {x,y}, a set containing x and y as its only elements.

4.Axiom of union: Every set has a union. That is, for any set x there is a set y whose elements are precisely the elements of the elements of x.

5.Axiom of infinity: There exists a set x, one of whose members is {}, such that whenever y is in x, so is y U {y}.

6.Axiom schema of separation (or subset axiom): Given any set y and any first order logic formula P(x), where x is a free variable, there is a subset of y containing precisely those elements x for which P(x) comes out true.

7.Axiom schema of replacement: Roughly, if the domain of a function is a set, its range is also a set. More precisely, let w be some set with typical member x, and let P be a binary predicate. Then if P(x,y) and P(x,z) imply y = z (in which case P is a functional predicate and a mapping), there exists a set containing the image of w under P.

8.Axiom of power set: Every set has a power set. That is, for any set x there exists a set y, such that the elements of y are precisely the subsets of x.

9.Axiom of regularity (or axiom of foundation): Every non-empty set x contains some element y such that x and y are disjoint sets.

10.Axiom of choice: (Zermelo's version) Given a set x of mutually disjoint nonempty sets, there is a set y (a choice set for x) containing exactly one element from each member of x.

Well-ordered sets

A well-ordered set is an ordered set in which every non-empty subset has a least element.

Von Neumann definition of ordinals

A set S is an ordinal if and only if S is totally ordered with respect to set containment and every element of S is also a subset of S.

(Here, "set containment" is another name for the subset relationship.) Such a set S is automatically well-ordered with respect to set containment. This relies on the axiom of well foundation: every nonempty set B has an element b which is disjoint from B.

Note that the natural numbers are ordinals by this definition. For instance, 2 is an element of 4 = {0, 1, 2, 3}, and 2 is equal to {0, 1} and so it is a subset of {0, 1, 2, 3}.

It can be shown by transfinite induction that every well-ordered set is order-isomorphic to exactly one of these ordinals.

Furthermore, the elements of every ordinal are ordinals themselves. Whenever you have two ordinals S and T, S is an element of T if and only if S is a proper subset of T, and moreover, either S is an element of T, or T is an element of S, or they are equal. So every set of ordinals is totally ordered. And in fact, much more is true: Every set of ordinals is well-ordered. This important result generalizes the fact that every set of natural numbers is well-ordered and it allows us to use transfinite induction liberally with ordinals.

Another consequence is that every ordinal S is a set having as elements precisely the ordinals smaller than S. This statement completely determines the set-theoretic structure of every ordinal in terms of other ordinals. It is used to prove many other useful results about ordinals. One example of these is an important characterization of the order relation between ordinals: every set of ordinals has a supremum, the ordinal obtained by taking the union of all the ordinals in the set (this union exists regardless of the set's size, by the axiom of union). Another example is the fact that the collection of all ordinals is not a set. Indeed, since every ordinal contains only other ordinals, it follows that every member of the collection of all ordinals is also its subset.

Borel sets

The Borel algebra in an arbitrary topological space is the smallest collection of subsets of the space that contains the open sets and is closed under countable unions and complementation. It can be shown that the Borel algebra is closed under countable intersections as well.

A short proof that the Borel algebra is well defined proceeds by showing that the entire powerset of the space is closed under complements and countable unions, and thus the Borel algebra is the intersection of all families of subsets of the space that have these closure properties. This proof does not give a simple procedure for determining whether a set is Borel. A motivation for the Borel hierarchy is to provide a more explicit characterization of the Borel sets.

Boldface hierarchy

[Source: (edited) Wikipedia]