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. The statement 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.
Notes
References "Introductory note to 1947 and 1964", Gregory H. Moore, pp. 154-175, in Kurt Gödel: Collected Works: Volume II: Publications 1938-1974, Kurt Gödel, eds. S. Feferman, John W. Dawson, Jr., Stephen C. Kleene, G. Moore, R. Solovay, and Jean van Heijenoort, eds., New York, Oxford: Oxford University Press, 1990, . "Sur les ensembles analytiques nuls", Nicolas Lusin, Fundamenta Mathematicae, 25 (1935), pp. 109-131, . "History of the Continuum in the 20th Century", Juris Steprāns, pp. 73-144, in Handbook of the History of Logic: Volume 6: Sets and Extensions in the Twentieth Century, eds. Dov M. Gabbay, Akihiro Kanamori, John Woods, Amsterdam, etc.: Elsevier, 2012, .