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 ${\Delta }_{\alpha }^0$ for $1\le \alpha <{\omega }_1^{\text{ck}}$. We characterize the reals that strongly relativize ${\Delta }_1^0$ subsets of $\mathbb{N}$ as the reals which are computably dominated. For $1\le \alpha <{\omega }_1^{\text{ck}}$, we show that there are continuum many reals which strongly relativize ${\Delta }_{\alpha }^0$ subsets of $\mathbb{N}$. We will also find non-trivial examples of reals which strongly relativize ${\Delta }_{\alpha }^0$ ($1\le \alpha <{\omega }_1^{\text{ck}}$) subsets of Baire space.

19 pages