### Messages

### Timetable

### Material

Some material on Determinacy, including the Gale-Stewart Theorem, which we will cover this week.

### Tasks

#### Projects

Here are some projects. You can do any of these with a partner:

1. Present the Souslin-Kleene Theorem: A set is borel iff it is analytic and co-analytic

2. Present the theorem: if there is a measurable cardinal then there are only countably many reals in L. (You should familiarise yourself with the concept of indiscernables.)

3. Present Shelah's result in the attached paper of S. Shelah called: Weakly Compact Cardinals: A Combinatorial Proof

4. (Tapio was offered and might choose this): Prove that \kappa weakly compact iff \kappa is Pi^1_1-indescribable iff [\kappa is inaccessible and L_{\kappa, \kappa} satisfies the weak compactness theorem.]

I will provide references to the standard literature if needed.

### WHAT WE COVERED IN THE LECTURES

First lecture of the new quarter: We reviewed facts about measurable cardinals: e.g. that they are weakly compact. We then introduced the Axiom of Determinacy (AD), proved that AD refutes the Axiom of Choice, also that AS implies countable choice.