Counterexamples to countable-section ${\Pi }_2^1$ uniformization and ${\Pi }_3^1$ separation

Abstract

We make use of a finite support product of the Jensen minimal ${\Pi }_2^1$ singleton forcing to define a model in which ${\Pi }_2^1$ Uniformization fails for a set with countable cross-sections. We also define appropriate submodels of the same model in which Separation fails for ${\Pi }_3^1$.

arXiv admin note: text overlap with arXiv:1408.3901

References in corpus (1)

• A definable ${\mathsf{E}}_0$-class containing no definable elements

Cited by in corpus (3)

• Definable ${\mathsf{E}}_0$ classes at arbitrary projective levels
• A countable definable set of reals containing no definable elements
• A model in which the Separation principle holds for a given effective projective Sigma-class