Countable OD sets of reals belong to the ground model

TL;DR

This paper proves that in various generic extensions, all countable ordinal-definable sets of reals are already present in the ground model, highlighting a form of definability preservation.

Contribution

It establishes that countable OD sets of reals in Cohen, Solovay-random, dominating, and Sacks extensions are contained in the ground model, extending understanding of definability in forcing extensions.

Findings

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

The same holds in Solovay-random, dominating, and Sacks extensions.

This result links definability properties across different generic extensions.

Abstract

It is true in the Cohen, Solovay-random, dominaning, and Sacks generic extension that every countable ordinal-definable set of reals belongs to to the ground universe