The Axiom of Real Determinacy and the Axiom of Real Blackwell Determinacy

Abstract

We show that the Axiom of Real Determinacy $\mathsf{AD}_{\mathbb{R}}$ and the Axiom of Real Blackwell Determinacy $\mathsf{Bl}\text{-}\mathsf{AD}_{\mathbb{R}}$ are equivalent in $\mathsf{ZF}$+$\mathsf{DC}$. This answers the question of L\"{o}we [15, Question 53]. While we do not know whether they are equivalent in $\mathsf{ZF}$+$\mathsf{AC}_{\omega} (\mathbb{R})$, we show that the theories $\mathsf{ZF}$+$\mathsf{AD}_{\mathbb{R}}$ and $\mathsf{ZF}$+$\mathsf{AC}_{\omega} (\mathbb{R})$+$\mathsf{Bl}\text{-}\mathsf{AD}_{\mathbb{R}}$ are equiconsistent.