Approximation of hyperarithmetic analysis by $\omega$-model reflection

Abstract

This paper presents two types of results related to hyperarithmetic analysis. First, we introduce new variants of the dependent choice axiom, namely $\mathrm{unique}~\Pi^1_0(\mathrm{resp.}~\Sigma^1_1)\text{-}\mathsf{DC}_0$ and $\mathrm{finite}~\Pi^1_0(\mathrm{resp.}~\Sigma^1_1)\text{-}\mathsf{DC}_0$. These variants imply $\mathsf{ACA}_0^+$ but do not imply $\Sigma^1_1\mathrm{~Induction}$. We also demonstrate that these variants belong to hyperarithmetic analysis and explore their implications with well-known theories in hyperarithmetic analysis. Second, we show that $\mathsf{RFN}^{-1}(\mathsf{ATR}_0)$, a class of theories defined using the $\omega$-model reflection axiom, approximates to some extent hyperarithmetic analysis, and investigate the similarities between this class and hyperarithmetic analysis.