The second continuum hypothesis, also called Luzin's hypothesis or Luzin's second continuum hypothesis, is the hypothesis that

2^=2^

. It is the negation of a weakened form,

2^<2^

, 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). It was discussed by

Nikolai Luzin Nikolai Nikolayevich Luzin (also spelled Lusin; rus, Никола́й Никола́евич Лу́зин, p=nʲɪkɐˈlaj nʲɪkɐˈlajɪvʲɪtɕ ˈluzʲɪn, a=Ru-Nikilai Nikilayevich Luzin.ogg; 9 December 1883 – 28 February 1950) was a Sov ...

in 1935, although he did not claim to be the first to postulate it. The statement

2^<2^

may also be called Luzin's hypothesis. The second continuum hypothesis is independent of

Zermelo–Fraenkel set theory In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes suc ...

with 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 ...

(ZFC): its truth is consistent with ZFC since it is true in

Cohen Cohen () is a surname of Jewish, Samaritan and Biblical origins (see: Kohen). It is a very common Jewish surname (the most common in Israel). Cohen is one of the four Samaritan last names that exist in the modern day. Many Jewish immigrants ente ...

's model of ZFC with the negation of the Continuum Hypothesis; its falsity is also consistent since it is contradicted by the

, which follows from

V=L The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as ''V'' = ''L''. The axiom, first investigated by Kurt Gödel, is inconsistent with the pr ...

. It is implied by

Martin's Axiom In the mathematical field of set theory, Martin's axiom, introduced by Donald A. Martin and Robert M. Solovay, is a statement that is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, but it is consi ...

together with the negation of the CH.

Notes

References

{{reflist, refs= {{Cite journal , last=Cohen , first=Paul J. , date=15 December 1963 , title=The independence of the Continuum Hypothesis, art I , journal=Proceedings of the National Academy of Sciences of the United States of America , volume=50 , issue=6 , pages=1143–1148 , doi=10.1073/pnas.50.6.1143 , pmid=16578557 , pmc=221287 , jstor=71858 , bibcode=1963PNAS...50.1143C , doi-access=free {{Cite journal , last=Cohen , first=Paul J. , date=15 January 1964 , title=The independence of the Continuum Hypothesis,

art Art is a diverse range of cultural activity centered around ''works'' utilizing creative or imaginative talents, which are expected to evoke a worthwhile experience, generally through an expression of emotional power, conceptual ideas, tec ...

nbsp;II , journal=Proceedings of the National Academy of Sciences of the United States of America , volume=51 , issue=1 , pages=105–110 , doi=10.1073/pnas.51.1.105 , pmid=16591132 , pmc=300611 , jstor=72252 , bibcode=1964PNAS...51..105C , doi-access=free {{SpringerEOM, title = Luzin hypothesis "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. "Sur les ensembles analytiques nuls"
Nicolas Lusin, ''Fundamenta Mathematicae'', 25 (1935), pp. 109-131, {{doi, 10.4064/fm-25-1-109-131. "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. Infinity Hypotheses Cardinal numbers