The second continuum hypothesis, also called Luzin's hypothesis or Luzin's second continuum hypothesis, is the hypothesis that . It is the negation of a weakened form, , 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 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'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.[2]
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