October 24
Bartosz Wcisło,
University of Gdańsk
Levels of projective determinacy and levels of dependent choice
Gitman, Friedman, and Kanovei described a construction of a model of ZF in which countable choice holds and $\Pi^1_2$-dependent choice for reals is not satisfied. We modify that construction to obtain a model in which (boldface) $\Pi^1_n$-determinacy holds, but which fails to satisfy (lightface) $\Pi^1_{n+2}$-${\rm DC}$ for reals. In particular, we show that no projective level of determinacy implies full ${\rm DC}_{\mathbb R}$. This is joint work with Sandra Müller.