OD elements of countable OD sets in the Solovay model

TL;DR

In the Solovay model, all countable ordinal-definable sets of sets of reals consist solely of ordinal-definable elements, highlighting a specific structural property of definability within this model.

Contribution

This paper proves a new property of countable OD sets of sets of reals in the Solovay model, showing they contain only OD elements, which was previously unknown.

Findings

Countable OD sets of sets of reals have only OD elements in the Solovay model.

The result clarifies the structure of definable sets in the Solovay model.

Supports the understanding of OD set properties in set theory.

Abstract

It is true in the Solovay model that every countable ordinal-definable set of sets of reals contains only ordinal-definable elements.