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THE 
${\bf{\Sigma }}_2^1$ COUNTERPARTS TO STATEMENTS THAT ARE EQUIVALENT TO THE CONTINUUM HYPOTHESIS
Published online by Cambridge University Press: 22 December 2015
Abstract
We consider natural 
${\rm{\Sigma }}_2^1$ definable analogues of many of the classical statements that have been shown to be equivalent to CH. It is shown that these 
${\rm{\Sigma }}_2^1$ analogues are equivalent to that all reals are constructible. We also prove two partition relations for 
${\rm{\Sigma }}_2^1$ colourings which hold precisely when there is a non-constructible real.
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