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) $\bm{\Pi^1_n}$-determinacy holds, but (lightface) $\Pi^1_{n+2}$-$\mathsf{DC}$ 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.