A model with fragments of projective determinacy and failures of $\mathsf{DC}$

Abstract

We describe a construction of a model of second order arithmetic in which (boldface) ${\mathbf{\Pi }}_{\mathbf{n}}^1$-determinacy holds, but (lightface) ${\Pi }_{n+2}^1$-$\mathsf{DC}$ fails, thus showing that no projective level of determinacy implies full ${\mathsf{DC}}_{\mathbb{R}}$. The construction builds upon the work of Gitman, Friedman, and Kanovei.

17 pages