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

• 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.