Friedman–Moschovakis coding lemma

Assume the axiom of determinacy. If $\lambda$ is a surjective image of the Baire space of sequences and, for every $\xi <\lambda$, the power set $\mathcal{P}(\xi )$ is a surjective image of that space, then $\mathcal{P}(\lambda )$ is also a surjective image of it.

For $A\subseteq \lambda$, the Friedman–Moschovakis coding game asks the players to present coherent local codes for initial segments $A\cap \xi$ while challenging each other at larger ordinals. The diagonal and boundedness argument rules out a winning strategy for Player I; a winning strategy for Player II determines at most one set $A$. Coding strategies by reals therefore gives a surjection from the Baire space of sequences onto $\mathcal{P}(\lambda )$.