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

Abstract

In this paper, we introduce a hierarchy dividing the set $\{\sigma \in {\Pi }_2^1:{\Pi }_1^1$-${\mathsf{CA}}_0⊢\sigma \}$. 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 ${\Sigma }_n^0$ classes of $[\mathbb{N}{]}^{\mathbb{N}}$ and determinacy for $({\Sigma }_1^0{)}_n$ classes of ${\mathbb{N}}^{\mathbb{N}}$.