On the $\Pi^1_2$ consequences of $\Pi^1_1$-$\mathsf{CA}_0$

Yudai Suzuki; Keita Yokoyama

On the $\Pi^1_2$ consequences of $\Pi^1_1$-$\mathsf{CA}_0$

Logic 2024-11-25 v2

Authors: Yudai Suzuki , Keita Yokoyama

Abstract

In this paper, we introduce a hierarchy dividing the set $\{\sigma \in \Pi^1_2 : \Pi^1_1$ - $\mathsf{CA}_0 \vdash \sigma\}$ . Then, we give some characterizations of this set using weaker variants of some principles equivalent to $\Pi^1_1$ - $\mathsf{CA}_0$ : leftmost path principle, Ramsey's theorem for $\Sigma^0_n$ classes of $[\mathbb{N}]^{\mathbb{N}}$ and determinacy for $(\Sigma^0_1)_n$ classes of $\mathbb{N}^{\mathbb{N}}$ .

Keywords

reverse mathematics $\Pi^1_1$-$\mathsf{CA}_0$ $\Pi^1_2$ consequences leftmost path principle Ramsey's theorem determinacy

Cite

@article{arxiv.2402.07136,
  title  = {On the $\Pi^1_2$ consequences of $\Pi^1_1$-$\mathsf{CA}_0$},
  author = {Yudai Suzuki and Keita Yokoyama},
  journal= {arXiv preprint arXiv:2402.07136},
  year   = {2024}
}

On the $\Pi^1_2$ consequences of $\Pi^1_1$-$\mathsf{CA}_0$

Abstract

Keywords

Cite

Related papers