The Limits of Determinacy in Higher-Order Arithmetic

Abstract

We prove level-by-level upper and lower bounds on the strength of determinacy for finite differences of sets in the hyperarithmetical hierarchy in terms of subsystems of finite-and transfinite-order arithmetic, extending the Montalbán-Shore theorem to each of the levels of the Borel hierarchy beyond the one they treated. We also prove equivalences between reflection principles for higher-order arithmetic and quantified determinacy axioms, answering two questions of Pacheco and Yokoyama.

Figures (12)

• Figure 1: Provability relations established in this article, for $1\leq \gamma \leq \omega_1^{\mathsf{CK}}$ and $1 \leq n < \omega$; none of the arrows can be reversed. Here, $\text{$\mathsf{KP}$}^\gamma_m$ denotes the theory $\text{$\mathsf{KP}$}$ augmented with the axioms "$\mathcal{P}^\gamma(\mathbb{N})$ exists" and "$\gamma$ is wellordered," as well as with the schemata of $\Delta_m$--separation and $\Sigma_m$--collection.
• Figure 2: A ${(\Pi^0_{\alpha})}_5$ set, where $A_2$ plays the same role as $A_2 \cap A_1$.
• Figure 3: A typical situation in the game of $\text{$\mathsf{KP}$}^{\gamma}_n$, for the case $H_1$.
• Figure 4: The game $G^{\gamma}_{n}$ for $n$ even.
• Figure 5: The game of theorem \ref{['notmPigammaDet']} for even $n$

Theorems & Definitions (99)

• Theorem 1.1: Montalbán and Shore
• Theorem 1.2: Montalbán and Shore
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: Montalbán and Shore MSCons
• Corollary 1.6
• proof
• Remark 1.7
• Theorem 1.8
• Definition 2.1: Effective Borel Hierarchy ($\mathsf{ATR}_0$)