Searching problems above arithmetical transfinite recursion

Abstract

We investigate some Weihrauch problems between ${\mathrm{ATR}}_2$ and $C_{{\omega }^{\omega }}$. We show that the fixed point theorem for monotone operators on the Cantor space (a weaker version of the Knaster-Tarski theorem) is not Weihrauch reducible to ${\mathrm{ATR}}_2$. Furthermore, we introduce the ω-model reflection ${\mathrm{ATR}}_2^{\mathrm{rfn}}$ of ${\mathrm{ATR}}_2$ and show that it is an upper bound for problems provable from the axiomatic system ${\mathrm{ATR}}_0$ which are of the form $\forall X\left(\theta \right(X)\to \exists Y\eta (X,Y\left)\right)$ with arithmetical formulas $\theta ,\eta$. We also show that Weihrauch degrees of relativized least fixed point theorems for monotone operators on the Cantor space form a linear hierarchy between ${\mathrm{ATR}}_2^{\mathrm{rfn}}$ and $C_{{\omega }^{\omega }}$.