April 18
Yudai Suzuki,
Tohoku University
On the $\Pi^1_2$ consequences of $\Pi^1_1$-$\mathsf{CA}_0$
(This is a joint work with Keita Yokoyama) Some recent studies in reverse mathematics focus on theorems that are stronger than $\mathsf{ATR}_0$ and weaker than $\Pi^1_1$-$\mathsf{CA}_0$. Especially, theorems represented by a $\Pi^1_2$ sentence and provable from $\Pi^1_1$-$\mathsf{CA}_0$ are investigated. In this talk, we will introduce a hierarchy slicing the set $\{\sigma \in \Pi^1_2: \Pi^1_1\text{-}\mathsf{CA}_0\}$. Then, we show that Ramsey's theorem for $\Sigma^0_2,\Sigma^0_3,\Sigma^0_4,\ldots$ sets are separated in a sense although all of them are equivalent to $\Pi^1_1$-$\mathsf{CA}_0$ by comparing this hierarchy and a suitable weaker variant of those theorems. We also show that a similar result holds for $(\Sigma^0_1)_n$ determinacy.