# Σ11 in every real in a Σ11 class of reals is Σ11

## Richard Shore

### Cornell University

We first prove a theorem about reals (subsets of **N**) and classes of reals: If a real *X* is Σ^{1}_{1} in every member *G* of a nonempty Σ^{1}_{1} class **K** of reals then *X* is itself Σ^{1}_{1}. We also explore the relationship between this theorem, various basis results in hyperarithmetic theory and omitting types theorems in ω-logic. We then prove the analog of our first theorem for classes of reals: If a class **A** of reals is Σ^{1}_{1} in every member of a nonempty Σ^{1}_{1} class **B** of reals then **A** is itself Σ^{1}_{1}. This is joint work with T. Slaman and L. Harrington .

Richard Shore is the Goldwin Smith Professor of Mathematics at Cornell University, and is a past president of the Association for Symbolic Logic. He received his doctorate from the Massachusetts Institute of Technology in 1972, under the supervision of Gerald Sacks. At Cornell he has directed 16 doctoral theses and mentored a dozen postdoctoral scholars.