Some algebraic equivalent forms of $\mathbb{R} \subseteq L$

TL;DR

This paper explores algebraic forms related to the Continuum Hypothesis within a definability framework, showing their equivalence to the statement that all reals are constructible.

Contribution

It introduces $oldsymbol{ ext{Σ}}^1_2$ definable versions of algebraic forms of the Continuum Hypothesis and proves their equivalence to the constructibility of all reals.

Findings

All considered algebraic forms are equivalent to 'all reals are constructible'

$oldsymbol{ ext{Σ}}^1_2$ definable counterparts mirror classical algebraic forms

The results connect definability with classical set-theoretic hypotheses

Abstract

We study $\Sigma^1_2$ definable counterparts for some algebraic equivalent forms of the Continuum Hypothesis. All turn out to be equivalent to "all reals are constructible".