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.
Categories: math.LO