The Σ¹₂ counterparts to statements that are equivalent to the Continuum Hypothesis

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We consider natural Σ¹₂ definable analogues of many of the classical statements that have been shown to be equivalent to CH. It is shown that these Σ¹₂ analogues are equivalent to that all reals are constructible. We also prove two partition relations for Σ¹₂ colourings which hold precisely when there is a non-constructible real.