Counterexamples to countable-section $\varPi^1_2$ uniformization and $\varPi^1_3$ separation

TL;DR

This paper constructs models in set theory where certain uniformization and separation properties fail at specific projective levels, using advanced forcing techniques.

Contribution

It introduces a novel forcing construction to demonstrate the failure of $oldsymbol{ m ext{Pi}^1_2}$ uniformization and $oldsymbol{ m ext{Pi}^1_3}$ separation in particular models.

Findings

$oldsymbol{ m ext{Pi}^1_2}$ uniformization fails for sets with countable cross-sections.

$oldsymbol{ m ext{Pi}^1_3}$ separation fails in certain submodels.

The method uses a finite support product of Jensen minimal $oldsymbol{ m ext{Pi}^1_2}$ singleton forcing.

Abstract

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