Second continuum hypothesis

The second continuum hypothesis, also called Luzin's hypothesis or Luzin's second continuum hypothesis, is the hypothesis that $2^{\aleph _{0}}=2^{\aleph _{1}}$ . It is the negation of a weakened form, $2^{\aleph _{0}}<2^{\aleph _{1}}$ , 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 $2^{\aleph _{0}}<2^{\aleph _{1}}$ 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

^ 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

^ Jump up to: ^a ^b "Sur les ensembles analytiques nuls", Nicolas Lusin, Fundamenta Mathematicae, 25 (1935), pp. 109-131, doi:10.4064/fm-25-1-109-131.
^ Jump up to: ^a ^b ^c "Luzin hypothesis", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
^ "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. Bibcode:1963PNAS...50.1143C. doi:10.1073/pnas.50.6.1143. JSTOR 71858. PMC 221287. PMID 16578557.
^ 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.

[2] 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

[L1935-1] Jump up to: ^a ^b "Sur les ensembles analytiques nuls", Nicolas Lusin, Fundamenta Mathematicae, 25 (1935), pp. 109-131, doi:10.4064/fm-25-1-109-131.

[EOM-3] Jump up to: ^a ^b ^c "Luzin hypothesis", Encyclopedia of Mathematics, EMS Press, 2001 [1994]

[G1990II-4] "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.

[S2012-5] "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.

[C1963-6] 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.

[C1964-7] 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.

[note 1]

[2]

[3]

[4]

[1]

[5]

[6]