Math 787 — Set Theory

Spring 2003

Instructor: Russell Miller

Time: TR 11:40-12:55

Room: MT 230

Math 787 is a course on Axiomatic Set Theory. It will be taught at an introductory level, not assuming any previous knowledge of the subject. Topics in this course include the axioms of Zermelo-Frankel (ZF) set theory; ordinals and cardinals; Von Neumann rank and reflection principles; absoluteness; inner models; infinitary combinatorics and Martin's Axiom; Suslin's Hypothesis; Godel's universe of constructible sets (L); and basic Cohen forcing. We will use these last two topics to prove the independence of the Axiom of Choice (AC) from ZF, and the independence of the Generalized Continuum Hypothesis (GCH) from ZFC (i.e. from ZF + AC). If time permits, we may also discuss iterated forcing and/or results on large cardinals.

The course will generally follow Kunen's book, which covers all the topics mentioned above except large cardinals. Other useful books on the subject are listed below. As Kunen's title suggests, the course will focus on consistency and independence results in the axiomatic development of set theory. Math 681, or any course covering predicate logic, is quite sufficient as a prerequisite. Indeed, the beginning of this course may overlap with topics taught in certain years in 681.

References

Kenneth Kunen, Set Theory: An Introduction to Independence Proofs (Amsterdam: Elsevier, 1980). (Available in paperback!)

Erdos, Hajnal, Mate, & Rado, Combinatorial Set Theory: Partition Relations for Cardinals (Amsterdam: North-Holland, 1984).

Hajnal & Hamburger, Set Theory, trans. Mate (Cambridge: Cambridge Univ. Press, 1999).

Last modified: April 7, 2003