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

Yudai Suzuki; Keita Yokoyama

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arXiv · 2402.07136

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

Yudai Suzuki·Keita Yokoyama

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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 $\mathbb{N}^{\mathbb{N}}$.

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BibTeX RIS

Yudai Suzuki, Keita Yokoyama. 2024-02-11. On the $\Pi^1_2$ consequences of $\Pi^1_1$-$\mathsf{CA}_0$. https://arxiv.org/abs/2402.07136

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