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

TL;DR

This paper constructs a model of second order arithmetic demonstrating that projective determinacy at a certain level does not imply the full Axiom of Dependent Choices for reals, highlighting limitations in the implications of determinacy.

Contribution

It introduces a novel model showing that projective determinacy does not entail full $ ext{DC}_{ ext{R}}$, extending previous work by Gitman, Friedman, and Kanovei.

Findings

Projective determinacy at a certain level does not imply full $ ext{DC}_{ ext{R}}$.

The construction builds on prior foundational work.

The model separates levels of determinacy from choice principles.

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.