Second continuum hypothesis

The second continuum hypothesis, also called Luzin's hypothesis or Luzin's second continuum hypothesis, is the hypothesis that $2^{{\mathrm{\aleph }}_0}=2^{{\mathrm{\aleph }}_1}$. It is the negation of a weakened form, $2^{{\mathrm{\aleph }}_0}<2^{{\mathrm{\aleph }}_1}$, 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.Template:RefnTemplate:RTemplate:RTemplate:RTemplate:R The statement $2^{{\mathrm{\aleph }}_0}<2^{{\mathrm{\aleph }}_1}$ may also be called Luzin's hypothesis.Template:R

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;Template:RTemplate:R its falsity is also consistent since it is contradicted by the continuum hypothesis, which follows from V=L. It is implied by Martin's axiom together with the negation of the CH.Template:R