The $ \mathbf{\Sigma}^1_2$ counterparts to statements that are equivalent to the Continuum Hypothesis

TL;DR

This paper explores $oldsymbol{ ext{Σ}}^1_2$ definable statements related to the Continuum Hypothesis, establishing their equivalence to all reals being constructible and analyzing partition relations tied to non-constructible reals.

Contribution

It introduces $oldsymbol{ ext{Σ}}^1_2$ analogues of classical CH statements and proves their equivalence to the constructibility of all reals, along with new partition relations.

Findings

$oldsymbol{ ext{Σ}}^1_2$ analogues are equivalent to all reals being constructible.

Partition relations for $oldsymbol{ ext{Σ}}^1_2$ colorings hold iff a non-constructible real exists.

The work links definability, constructibility, and combinatorial properties in set theory.

Abstract

We consider natural $\Sigma^1_2$ definable analogues of many of the classical statements that have been shown to be equivalent to CH. It is shown that these $\Sigma^1_2$ analogues are equivalent to that all reals are constructible. We also prove two partition relations for $\Sigma^1_2$ colourings which hold precisely when there is a non-constructible real.