Strongly relativizing reals

Abstract

For a real $f\in \mathbb{N}^\mathbb{N}$, it is in general not the case that every set which is both $\Sigma^1_1(f)$ and $\Pi^1_1(f)$ is the $f$-section of a $\Delta^1_1$ set. However, there are reals $f$ for which this, in fact, does happen; we say that such a real strongly relativizes $\Delta^1_1$. In this paper, we will prove that the reals which strongly relativize $\Delta^1_1$ are exactly the hyperlow reals, i.e., the $f\in \mathbb{N}^\mathbb{N}$ with $\omega_1^f=\omega_1^{\text{ck}}$. We will also study reals that strongly relativize the classes $\Delta^0_\alpha$ for $1\leq \alpha<\omega_1^{\text{ck}}$. We characterize the reals that strongly relativize $\Delta^0_1$ subsets of $\mathbb{N}$ as the reals which are computably dominated. For $1\leq \alpha<\omega_1^{\text{ck}}$, we show that there are continuum many reals which strongly relativize $\Delta^0_\alpha$ subsets of $\mathbb{N}$. We will also find non-trivial examples of reals which strongly relativize $\Delta^0_{\alpha}$ ($1\leq \alpha<\omega_1^{\text{ck}}$) subsets of Baire space.