A cardinal $\delta$ is a Woodin cardinal if for every function $f: \delta \to \delta$, there exists an elementary embedding $j: V \to M$ with critical point $\kappa < \delta$ such that $j(f)(\kappa) = f(\kappa)$... More precisely, $\delta$ is Woodin if for every $A \subseteq V_\delta$, there exists a $\kappa < \delta$ that is $< \delta$-$A$-strong: for all $\lambda < \delta$, there exists $j: V \to M$ with critical point $\kappa$, $j(\kappa) > \lambda$, $V_\lambda \subseteq M$, and $A \cap V_\lambda = j(A) \cap V_\lambda$.

A set $A \subseteq \omega^\omega$ (a subset of Baire space) is determined if one of the two players in the Gale-Stewart game $G_A$ has a winning strategy. In the game, players I and II alternately choose natural numbers $a_0, a_1, a_2, \ldots$, and player I wins if the resulting sequence $(a_0, a_1, \ldots) \in A$.

Projective determinacy (PD) asserts that every projective subset of $\omega^\omega$ is determined. $\mathrm{AD}$ (axiom of determinacy) asserts that every subset of $\omega^\omega$ is determined (this contradicts AC but is consistent with ZF).