The second continuum hypothesis, also called Luzin's hypothesis or Luzin's second continuum hypothesis, is the hypothesis that 2 ℵ 0 = 2 ℵ 1 {\displaystyle 2^{\aleph _{0}}=2^{\aleph _{1}}} . It is the negation of a weakened form, 2 ℵ 0 < 2 ℵ 1 {\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. The statement 2 ℵ 0 < 2 ℵ 1 {\displaystyle 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 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.
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