The term weak continuum hypothesis can be used to refer to the hypothesis that

2^<2^

, which is the negation of the second continuum hypothesis. It is equivalent to a weak form of ◊ on

\aleph_1

. F. Burton Jones proved that if it is true, then every separable normal Moore space is

metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \tau) is said to be metrizable if there is a metric d : X \times X \to , \infty) suc ...

. ''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 A cantor or chanter is a person who leads people in singing or sometimes in prayer. Cantor as a profession generally refers to those leading a Jewish congregation, although it also applies to the lead singer or choir director in Christian contexts. ...

's original form of the

Continuum Hypothesis In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this ...

(CH). Given the

Axiom of Choice In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...

, it is equivalent to the usual form of CH, that

2^=\aleph_1

References

{{reflist, refs= {{citation , last1=Coskey , first1=Samuel , last2=Ilijas , first2=Farah , title=Automorphisms of corona algebras, and group cohomology , journal=

Transactions of the American Mathematical Society The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must ...

, volume=366 , issue=7 , date=July 2014 , pages=3611–3630 , doi=10.1090/S0002-9947-2014-06146-1 , doi-access=free, arxiv=1204.4839 . {{citation , last1=Devlin , first1=Keith J. , authorlink1=Keith Devlin , last2=Shelah , first2=Saharon , authorlink2=Saharon Shelah , url=https://shelah.logic.at/files/95718/65.pdf , title=A weak version of ◊ which follows from

2^{\aleph_0}<2^{\aleph_1}

, journal=

Israel Journal of Mathematics '' Israel Journal of Mathematics'' is a peer-reviewed mathematics journal published by the Hebrew University of Jerusalem ( Magnes Press). History Founded in 1963, as a continuation of the ''Bulletin of the Research Council of Israel'' (Section ...

, volume=29 , issue=2–3 , date=1978 , pages=239–247 , mr=0469756 , doi=10.1007/BF02762012. {{citation , last1=Garti , first1=Shimon , title=Weak diamond and Galvin's property , arxiv=1603.06684 , journal=Periodica Mathematica Hungarica , volume=74 , issue=1 , pages=128–136 , date=2017 , doi=10.1007/s10998-016-0153-0. "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. {{citation , last = Jones , first = F. B. , authorlink = F. Burton Jones , doi = 10.1090/S0002-9904-1937-06622-5 , issue = 10 , journal =

Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. ...

, mr = 1563615 , pages = 671–677 , title = Concerning normal and completely normal spaces , volume = 43 , year = 1937, doi-access = free . "Topics in Set Theory"
lecture notes from lectures of O. Kolman, Michaelmas Term 2012, University of Cambridge. "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. "The Continuum Problem"

John Stillwell John Colin Stillwell (born 1942) is an Australian mathematician on the faculties of the University of San Francisco and Monash University. Biography He was born in Melbourne, Australia and lived there until he went to the Massachusetts Instit ...

, ''

The American Mathematical Monthly ''The American Mathematical Monthly'' is a peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposito ...

'', 109, #3 (March 2002), pp. 286-297, {{doi, 10.2307/2695360. Infinity Hypotheses Cardinal numbers