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

TL;DR

The paper proves that $\mathsf{AD}_{\mathbb{R}}$ and $\mathsf{Bl}\text{-}\mathsf{AD}_{\mathbb{R}}$ are equivalent in $\mathsf{ZF}$+$\mathsf{DC}$, and shows equiconsistency with $\mathsf{ZF}$+$\mathsf{AC}_{\omega}(\mathbb{R})$+$\mathsf{Bl}\text{-}\mathsf{AD}_{\mathbb{R}}$. It develops a fine normal measure on $\wp_{\omega_1}(\mathbb{R})$, proves that $\mathsf{Bl}\text{-}\mathsf{AD}_{\mathbb{R}}$ implies the Baire property and that every set of reals is $\infty$-Borel (indeed strongly $\infty$-Borel under $\mathsf{DC}$), and then uses infinitary coding to derive equivalence with $\mathsf{AD}_{\mathbb{R}}$ through Suslin-uniformization arguments. The work also establishes equiconsistency results via inner-model techniques and Ultrapower/Vopěnka-type constructions, and ends with open questions about determinacy under weaker choice assumptions. Altogether, it clarifies the relationship between strong determinacy axioms and their Blackwell-Determinacy analogues in weakened set theories.

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 Löwe [15, Question 53]. While we do not know whether they are equivalent in $\mathsf{ZF}$+$\mathsf{AC}_ω (\mathbb{R})$, we show that the theories $\mathsf{ZF}$+$\mathsf{AD}_{\mathbb{R}}$ and $\mathsf{ZF}$+$\mathsf{AC}_ω (\mathbb{R})$+$\mathsf{Bl}\text{-}\mathsf{AD}_{\mathbb{R}}$ are equiconsistent.

Theorems & Definitions (130)

• Theorem 1: Theorem \ref{['almost theorem']}
• Theorem 2: Theorem \ref{['equicon']}
• Theorem 3: Folklore?
• proof
• Theorem 4: Folklore?
• proof
• Lemma 5
• proof
• Definition 6
• Theorem 7