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 $Σ^1_1(f)$ and $Π^1_1(f)$ is the $f$-section of a $Δ^1_1$ set. However, there are reals $f$ for which this, in fact, does happen; we say that such a real strongly relativizes $Δ^1_1$. In this paper, we will prove that the reals which strongly relativize $Δ^1_1$ are exactly the hyperlow reals, i.e., the $f\in \mathbb{N}^\mathbb{N}$ with $ω_1^f=ω_1^{\text{ck}}$. We will also study reals that strongly relativize the classes $Δ^0_α$ for $1\leq α<ω_1^{\text{ck}}$. We characterize the reals that strongly relativize $Δ^0_1$ subsets of $\mathbb{N}$ as the reals which are computably dominated. For $1\leq α<ω_1^{\text{ck}}$, we show that there are continuum many reals which strongly relativize $Δ^0_α$ subsets of $\mathbb{N}$. We will also find non-trivial examples of reals which strongly relativize $Δ^0_α$ ($1\leq α<ω_1^{\text{ck}}$) subsets of Baire space.