In Cohen generic extension, every countable OD set of reals belongs to the ground model

TL;DR

In Cohen generic extensions of L, all countable ordinal-definable sets of reals are contained within the original universe, demonstrating a preservation property of definable sets under forcing.

Contribution

This paper proves that in Cohen generic extensions of L, every countable OD set of reals remains in the ground model, extending understanding of definability and forcing.

Findings

Countable OD sets of reals in Cohen extensions are in the ground model.

The result holds specifically in Cohen generic extensions of L.

It highlights a stability property of OD sets under Cohen forcing.

Abstract

It is true in the Cohen generic extension of L, the constructible universe, that every countable ordinal-definable set of reals belongs to L.