Second continuum hypothesis

The second continuum hypothesis, also called Luzin's hypothesis or Luzin's second continuum hypothesis, is the hypothesis that ${\displaystyle 2^{\aleph _{0}}=2^{\aleph _{1}}}$. It is the negation of a weakened form, ${\displaystyle 2^{\aleph _{0}}<2^{\aleph _{1}}}$, 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.^{[note 1][2][3]: 157, 171 [4]: §3 [1]: 130–131} The statement ${\displaystyle 2^{\aleph _{0}}<2^{\aleph _{1}}}$ may also be called Luzin's hypothesis.^[2]

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;^{[5][6]: 109–110} 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.^[2]

Notes

1. He didn't know who was the first: "Nous ne chercherons pas à donner le nom de l'auteur qui a conçu le premier la sériuse possibilité d'une telle hypothèse du continu..." ^[1]: 130