second continuum hypothesis
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.{{refn|group=note|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..." {{r|L1935|p=130}} }}{{r|EOM}}{{r|G1990II|pages=157, 171}}{{r|S2012|at=§3}}{{r|L1935|pages=130-131}} The statement may also be called Luzin's hypothesis.{{r|EOM}}
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;{{r|C1963}}{{r|C1964|pages=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.{{r|EOM}}
Notes
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References
{{reflist|refs=
{{Cite journal
|last=Cohen |first=Paul J.
|date=15 December 1963
|title=The independence of the Continuum Hypothesis, [part I]
|journal=Proceedings of the National Academy of Sciences of the United States of America
|volume=50 |issue=6 |pages=1143–1148
|doi=10.1073/pnas.50.6.1143 |pmid=16578557
|pmc=221287 |jstor=71858 |bibcode=1963PNAS...50.1143C
|doi-access=free
}}
{{Cite journal
|last=Cohen |first=Paul J.
|date=15 January 1964
|title=The independence of the Continuum Hypothesis, [part] II
|journal=Proceedings of the National Academy of Sciences of the United States of America
|volume=51 |issue=1 |pages=105–110
|doi=10.1073/pnas.51.1.105 |pmid=16591132
|pmc=300611 |jstor=72252 |bibcode=1964PNAS...51..105C
|doi-access=free
}}
{{SpringerEOM|title = Luzin hypothesis}}
"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, {{ISBN|0-19-503972-6}}.
"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, {{ISBN|978-0-444-51621-3}}.
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