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.[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 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.[2]

Notes

  1. ^ 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..." [1]: 130 

References

  1. ^ a b "Sur les ensembles analytiques nuls", Nicolas Lusin, Fundamenta Mathematicae, 25 (1935), pp. 109-131, doi:10.4064/fm-25-1-109-131.
  2. ^ a b c "Luzin hypothesis", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  3. ^ "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.
  4. ^ "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.
  5. ^ 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. Bibcode:1963PNAS...50.1143C. doi:10.1073/pnas.50.6.1143. JSTOR 71858. PMC 221287. PMID 16578557.
  6. ^ 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. Bibcode:1964PNAS...51..105C. doi:10.1073/pnas.51.1.105. JSTOR 72252. PMC 300611. PMID 16591132.