Infinite Games in Set Theory: Supercompactness of Aleph 1 and Aleph 2

We consider the properties of cardinals under various versions of the Axiom of Determinacy (AD), a possible set theoretic axiom that conflicts with the full Axiom of Choice. AD asserts the determinacy of certain infinite games which can be used to explore the complexity of subsets of a base set, typically taken to be the real numbers. Under AD some small (infinite) cardinals exhibit properties such as measurability and supercompactness which are normally associated only with very large cardinals. This essay explores some such results.