Weak axioms of determinacy and subsystems of analysis II (∑02 games)

AbstractIn [10], we have shown that the statement that all ∑11 partitions are Ramsey is deducible over ATR0 from the axiom of ∑11 monotone inductive definition,but the reversal needs П11-CA0 rather than ATR0. By contrast, we show in this paper that the statement that all ∑02 games are determinate is also deducible over ATR0 from the axiom of ∑11 monotone inductive definition, but the reversal is provable even in ACA0. These results illuminate the substantial differences among lightface theorems which can not be observed in boldface