MATH 7870: Set Theory (Spring 2011)

Instructor: Justin Moore

The goal of this course is to teach forcing and to develop the necessary set theory and motivating examples along the way. The student is assumed to have the level of exposure to set theory which is typically provided by a core course in analysis or topology. (Students will benefit from some exposure to the notions of measure and category, set cardinality, and Zorn's Lemma proofs.)

The course will be based around several case studies of forcings which illustrate the different effects one can produce using forcing and the different techniques which appear in proofs concerning forcing: collapsing cardinals, the countable chain condition, countable closure, fusion arguments, pure extension, reflection, product forcing, and finite iterated forcing.

In addition, the following set theoretic concepts will be introduced along the way: the ordinals and the cummulative hierarchy, stationary sets, Aronszajn trees, ultrafilters, gaps, and large cardinals (inaccessible, weakly compact, measurable).

We will cover the proofs of the independence of the Continuum Hypothesis, Solovay's model in which all subsets of the real line are Lebesgue measurable, as well as give an introduction to forcing axioms such as Martin's Axiom.

There will not be an official text for the course, but Kunen's Set Theory (North-Holland) contains much of the material. Students may find this as useful supplemental reading.