A countable definable set of reals containing no definable elements

Abstract

We make use of a finite support product of Jensen forcing to define a model in which there is a countable non-empty lightface ${\Pi }_2^1$ set of reals containing no ordinal-definable real.

Revised version. The revision includes an updated proof of Lemma 4.5 (the density-preservation lemma for the product)

References in corpus (2)

• Non-uniformizable sets with countable cross-sections on a given level of the projective hierarchy
• Counterexamples to countable-section ${\Pi }_2^1$ uniformization and ${\Pi }_3^1$ separation

Cited by in corpus (4)

• Definable ${\mathsf{E}}_0$ classes at arbitrary projective levels
• Countable OD sets of reals belong to the ground model
• In Cohen generic extension, every countable OD set of reals belongs to the ground model
• OD elements of countable OD sets in the Solovay model