The axiom of determinacy implies dependent choice in mice

摘要

We show that the Axiom of Dependent Choice, DC, holds in countably iterable, passive premice M constructed over their reals which satisfy the Axiom of Determinacy, AD, in a ZF + DC_(RM) background universe. This generalizes an argument of Kechris for L(R) using Steel's analysis of scales in mice. In particular, we show that for any n ≤ ω and any countable set of reals A so that M_n(A)∩R = A and M_n(A) ? AD, we have that M_n(A) ? DC.