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

Abstract

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

arXiv commentSeveral minor corrections throughout. Finally submitted