Second continuum hypothesis

The second continuum hypothesis, also called Luzin's hypothesis or Luzin's second continuum hypothesis, is the hypothesis that $2^{\aleph _{0}}=2^{\aleph _{1}}$. It is the negation of a weakened form, $2^{\aleph _{0}}<2^{\aleph _{1}}$, of the Continuum Hypothesis (CH). It was discussed by Nikolai Luzin in 1935, although he did not claim to be the first to postulate it. The statement $2^{\aleph _{0}}<2^{\aleph _{1}}$ may also be called Luzin's hypothesis.

The second continuum hypothesis is independent of Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC): its truth is consistent with ZFC since it is true in Cohen's model of ZFC with the negation of the Continuum Hypothesis; its falsity is also consistent since it's contradicted by the Continuum Hypothesis, which follows from V=L. It is implied by Martin's Axiom together with the negation of the CH.