# Math 512: Set Theory II

# Spring 1999

### Instructor David Marker

- Office: 411 SEO
- Office Phone: (312) 996-3069
- e-mail: marker@math.uic.edu

## Description

This course will be a continuation of Set Theory I. The topics covered will
include
- Review of Goedel's consturutible universe L.
- Forcing: Cohen introduced forcing to prove the independence of the
Continuum Hypothesis and the Axiom of Choice. Forcing is a powerful method for
proving independence results.
- Large Cardinals: One way to extend the axioms of ZFC is to claim the
existence of large cardinals that can not be proved to exist in ZFC. We will
look at measurable cardinal and examine their effect on the real numbers and
the constructible universe.
- Determinacy: Let X be a set of infinite sequences of natural numbers.
Consider a game where two players take turns playing natural numbers. Player I
wins the game if the sequence they play is in X, otherwise II wins. We say the
game is determined if one of the two players has a winning strategy.
The axiom of determinacy (AD) asserts that every game is determined. In ZFC
we can prove that all Borel games are determined. We consider AD its effect on
the reals and connections to large cardinals.

### References

- K. Kunen, Set Theory: An Introduction to Independence Proofs, North-Holland,
1980.
- T. Jech, Set Theory, Springer-Verlag, 1997.

Kunen is a good reference for forcing. Jech is the best reference for large
cardinals and determinacy.

Go back to Dave's Home Page