$\Sigma _{1}^{1}$ in Every Real in a $\Sigma _{1}^{1}$ Class of Reals Is $\Sigma _{1}^{1}$

Abstract

We first prove a theorem about reals (subsets of $\mathbb {N}$) and classes of reals: If a real X is $\Sigma _{1}^{1}$ in every member G of a nonempty $\Sigma _{1}^{1}$ class $\mathcal {K}$ of reals then X is itself $\Sigma _{1}^{1}$. We also explore the relationship between this theorem, various basis results in hyperarithmetic theory and omitting types theorems in $\omega$-logic. We then prove the analog of our first theorem for classes of reals: If a class $\mathcal {A}$ of reals is $\Sigma _{1}^{1}$ in every member of a nonempty $\Sigma _{1}^{1}$ class $\mathcal {B}$ of reals then $\mathcal {A}$ is itself $\Sigma _{1}^{1}$.

R.A. Shore—Partially supported by NSF Grant DMS-1161175. The last two authors began their work on this paper at a workshop of the Institute for Mathematical Sciences of the National University of Singapore which also partially supported them.

T.A. Slaman—Partially supported by NSF Grant DMS-1301659 and by the Institute for Mathematical Sciences of the National University of Singapore.