Second continuum hypothesis
The second continuum hypothesis, also called Luzin's hypothesis or Luzin's second continuum hypothesis, is the hypothesis that [math]\displaystyle{ 2^{\aleph_0}=2^{\aleph_1} }[/math]. It is the negation of a weakened form, [math]\displaystyle{ 2^{\aleph_0}\lt 2^{\aleph_1} }[/math], 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][1]:130-131 The statement [math]\displaystyle{ 2^{\aleph_0}\lt 2^{\aleph_1} }[/math] 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]
Notes
References
- ↑ 1.0 1.1 "Sur les ensembles analytiques nuls", Nicolas Lusin, Fundamenta Mathematicae, 25 (1935), pp. 109-131, doi:10.4064/fm-25-1-109-131.
- ↑ 2.0 2.1 2.2 Hazewinkel, Michiel, ed. (2001), "Luzin hypothesis", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page
- ↑ "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.
- ↑ Cohen, Paul J. (15 December 1963). "The independence of the Continuum Hypothesis, [part I"]. Proceedings of the National Academy of Sciences of the United States of America 50 (6): 1143–1148. doi:10.1073/pnas.50.6.1143. PMID 16578557. Bibcode: 1963PNAS...50.1143C.
- ↑ Cohen, Paul J. (15 January 1964). "The independence of the Continuum Hypothesis, [part II"]. Proceedings of the National Academy of Sciences of the United States of America 51 (1): 105–110. doi:10.1073/pnas.51.1.105. PMID 16591132. Bibcode: 1964PNAS...51..105C.
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