Set Theory without the Axiom of Choice

The first part of the course will discuss various consequences of AD, the Axiom of Determinacy. AD is inconsistent with AC. On the other hand, it implies many good properties hold for sets o real numbers. For example, AD implies LM (every set of reals is Lebesgue measurable), BP (every set of reals has the property of Baire), and P (every uncountable set of reals contains a homeomorphic copy of the Cantor set). In addition, it is connected with the theory of large cardinals, e.g., AD implies that the first uncountable cardinal, omega-one, is a measurable cardinal. For the second part of the course, we will cover Solovay's model of LM+BP+P. For the last part, we will present some models of set theory in which the axiom of choice fails as badly as we can conceive possible. For example, the Feferman-Levy model in which the real line is the countable union of countable sets.

These are lecture notes which I will update weekly:
ac.tex .. ac.pdf