How unprovable is Rabin’s decidability theorem?

We study the strength of set-theoretic axioms needed to prove Ra-bin’s theorem on the decidability of the MSO theory of the infinite binary tree. We first show that over the second-order arithmetic theory ACA0, the complementation theorem for nondeterministic tree automata is equivalent to a statement expressing the determinacy of all Gale-Stewart games given by Bool $\left( {\sum {_2^0} } \right)$ sets. It follows that the complementation theorem is provable from $\Pi _3^1$- but not $\Delta _3^1$ -comprehension.We then use results due to MedSalem-Tanaka, Möllerfeld and Heinatsch-Möllerfeld to prove that•the complementation theorem for non-deterministic tree automata,•the decidability of the $\Pi _3^1$ fragment of MSO on the infinite binary tree,•the positional determinacy of parity games, and•the determinacy of Bool$\left( {\sum {_2^0} } \right)$ Gale-Stewart gamesare all equivalent over $\Pi _3^1$-comprehension. It follows in particular that Rabin’s decidability theorem is not provable from $\Delta _3^1$-comprehension.