150.422 Axiomatic Set Theory

We will study the axiom system known as Zermelo-Fraenkel (ZF) set theory. There are various propositions that are of interest because we would like to know if either they or their negations are theorems of ZF. To show that a proposition is a theorem, one should find a proof of it from the axioms. To show that it is not a theorem requires more than just not being able to find a proof. The techniques involve are a major focus of the course.

One of the propositions of interest is the Axiom of Choice (AC), which states that, given a family F of non-empty sets, there is a choice function on F that picks out a single element from each set in F. This turns out to be equivalent to the statement that any set can be well-ordered, where by a well-ordering of a set we mean a strict linear ordering such that every non-empty subset has a least element. For example, the natural numbers come well-ordered but not the integers, rationals, or reals. It is straightforward to show that the integers and rationals, respectively, admit well-orderings. But the case of the reals is not so evident. Are we just lacking a proof, or is this evidence that the negation of AC is a theorem of ZF?

Another proposition of interest is the Continuum Hypothesis (CH). Georg Cantor pioneered the comparison in size (cardinality) of infinite sets (in addition to finite ones). Two sets have the same cardinality if there is a 1-1 correspondence between them. A set is strictly larger than another just in case the latter can be put into 1-1 correspondence with a subset of the former but not vice-versa. Cantor proved that the set of real numbers is strictly larger in this sense than the set of natural numbers. But is there a set of intermediate size? Cantor thought not, but couldn't prove it. The proposition that there is no intermediate set is the Continuum Hypothesis. If one could construct in ZF a set of intermediate size, the negation of CH would be a theorem. It would seem that either CH or the negation of CH should be a theorem.

During the course of the semester, we will learn to do the following.

• Argue that all of mathematics can be done in ZF, or in ZFC, which is ZF with the axiom of choice (AC) assuming AC does not already follow from ZF.
• Resolve the set-theoretic paradoxes (Russell, Burali-Forti, Cantor) and prove fundamental theorems within ZF or ZFC.
• Construct the hierarchy of transfinite ordinals and cardinals using the axioms of ZF or ZFC.
• Construct the hierarchy of well-founded sets.
• Learn about infinitary combinatorics.
• Master the concept of interpreting one theory in another.
• Develop from that the idea of relative consistency.
• Use relative consistency to establish various independence results.