Applications of infinity-Borel codes to definability and definable cardinals

Abstract

Woodin introduced an extension of the axiom of determinacy, AD, called AD+ which includes an assertion that all sets of reals have an ∞-Borel code. An ∞-Borel code is a pair (φ, S) where φ is a formula and S is a set of ordinals which provides a highly absolute definition for a set of reals. This paper will use AD+ and ∞-Borel codes to establish a property of ordinal definability analogous to a property for Σ11 shown by Harrington–Shore–Slaman (2017). Under AD+, the paper will also use ∞-Borel codes to explore the cardinality of sets below P(ω1) which Woodin (2006) began investigating under AD\BbbR and DC. The following summarizes the main results.