Weak continuum hypothesis

The term weak continuum hypothesis can be used to refer to the hypothesis that $2^{\aleph _{0}}<2^{\aleph _{1}}$ , which is the negation of the second continuum hypothesis.^[1]^: 80^[2]^{: Lecture 7}^[3]^: 3616 It is equivalent to a weak form of ◊ on $\aleph _{1}$ .^[4]^: 2^[5] F. Burton Jones proved that if it is true, then every separable normal Moore space is metrizable.^[6]^{: Theorem 5}

Weak continuum hypothesis may also refer to the assertion that every uncountable set of real numbers can be placed in bijective correspondence with the set of all reals. This second assertion was Cantor's original form of the Continuum Hypothesis (CH). Given the Axiom of Choice, it is equivalent to the usual form of CH, that $2^{\aleph _{0}}=\aleph _{1}$ .^[7]^: 155^[8]^: 289

References

^ "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.
^ "Topics in Set Theory", lecture notes from lectures of O. Kolman, Michaelmas Term 2012, University of Cambridge.
^ Coskey, Samuel; Ilijas, Farah (July 2014), "Automorphisms of corona algebras, and group cohomology", Transactions of the American Mathematical Society, 366 (7): 3611–3630, arXiv:1204.4839, doi:10.1090/S0002-9947-2014-06146-1.
^ Garti, Shimon (2017), "Weak diamond and Galvin's property", Periodica Mathematica Hungarica, 74 (1): 128–136, arXiv:1603.06684, doi:10.1007/s10998-016-0153-0.
^ Devlin, Keith J.; Shelah, Saharon (1978), "A weak version of ◊ which follows from $2^{\aleph _{0}}<2^{\aleph _{1}}$ " (PDF), Israel Journal of Mathematics, 29 (2–3): 239–247, doi:10.1007/BF02762012, MR 0469756.
^ Jones, F. B. (1937), "Concerning normal and completely normal spaces", Bulletin of the American Mathematical Society, 43 (10): 671–677, doi:10.1090/S0002-9904-1937-06622-5, MR 1563615.
^ "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.
^ "The Continuum Problem", John Stillwell, The American Mathematical Monthly, 109, #3 (March 2002), pp. 286-297, doi:10.2307/2695360.

[S2012-1] "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.

[K2012-2] "Topics in Set Theory", lecture notes from lectures of O. Kolman, Michaelmas Term 2012, University of Cambridge.

[CF2014-3] Coskey, Samuel; Ilijas, Farah (July 2014), "Automorphisms of corona algebras, and group cohomology", Transactions of the American Mathematical Society, 366 (7): 3611–3630, arXiv:1204.4839, doi:10.1090/S0002-9947-2014-06146-1.

[G2016-4] Garti, Shimon (2017), "Weak diamond and Galvin's property", Periodica Mathematica Hungarica, 74 (1): 128–136, arXiv:1603.06684, doi:10.1007/s10998-016-0153-0.

[DS1978-5] Devlin, Keith J.; Shelah, Saharon (1978), "A weak version of ◊ which follows from $2^{\aleph _{0}}<2^{\aleph _{1}}$ " (PDF), Israel Journal of Mathematics, 29 (2–3): 239–247, doi:10.1007/BF02762012, MR 0469756.

[J1937-6] Jones, F. B. (1937), "Concerning normal and completely normal spaces", Bulletin of the American Mathematical Society, 43 (10): 671–677, doi:10.1090/S0002-9904-1937-06622-5, MR 1563615.

[G1990II-7] "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.

[S2002-8] "The Continuum Problem", John Stillwell, The American Mathematical Monthly, 109, #3 (March 2002), pp. 286-297, doi:10.2307/2695360.

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