|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{About|the hypothesis in set theory|the assumption in fluid mechanics|Fluid mechanics}}
| | Hi and welcome. My name is Zack nevertheless I don't definitely like remaining referred to as like that. My career is an data officer. I at the moment stay in New Hampshire and my relatives loves it. To ride horses is the thing I appreciate most. Go to my web site to obtain out extra: http://netmax.isa.ru/netmax/view_profile.php?userid=641978<br><br>My page :: [http://netmax.isa.ru/netmax/view_profile.php?userid=641978 air max para mujer] |
| In [[mathematics]], the '''continuum hypothesis''' (abbreviated '''CH''') is a [[hypothesis]], advanced by [[Georg Cantor]] in 1878, about the possible sizes of [[infinite set]]s. It states:
| |
| :There is no set whose [[cardinality]] is strictly between that of the [[integer]]s and the [[real number]]s.
| |
| Establishing the truth or falsehood of this hypothesis is the first of [[Hilbert's problems|Hilbert's 23 problems]] presented in the year 1900. In 1963, [[Paul Cohen (mathematician)|Paul Cohen]] proved that the hypothesis is [[independence (mathematical logic)|independent]] of the axioms of [[Zermelo–Fraenkel set theory with the axiom of choice]] (ZFC), a standard foundation of mathematics, complementing earlier work by [[Kurt Gödel]] in 1940. Such independence means that ZFC can be augmented by either CH or its negation ¬CH, in both cases producing a system of axioms that is consistent if and only if ZFC is consistent.
| |
| Cohen was awarded the [[Fields Medal]] in 1966 for his proof.
| |
| | |
| The name of the hypothesis comes from the term ''[[continuum (set theory)|the continuum]]'' for the real numbers.
| |
| | |
| ==Cardinality of infinite sets==
| |
| {{Main|Cardinal number}}
| |
| | |
| Two sets are said to have the same ''[[cardinality]]'' or ''[[cardinal number]]'' if there exists a [[bijection]] (a one-to-one correspondence) between them. Intuitively, for two sets ''S'' and ''T'' to have the same cardinality means that it is possible to "pair off" elements of ''S'' with elements of ''T'' in such a fashion that every element of ''S'' is paired off with exactly one element of ''T'' and vice versa. Hence, the set {banana, apple, pear} has the same cardinality as {yellow, red, green}.
| |
| | |
| With infinite sets such as the set of [[integer]]s or [[rational number]]s, this becomes more complicated to demonstrate. The rational numbers seemingly form a counterexample to the continuum hypothesis: the integers form a proper subset of the rationals, which themselves form a proper subset of the reals, so intuitively, there are more rational numbers than integers, and more real numbers than rational numbers. However, this intuitive analysis does not take account of the fact that all three sets are [[infinite set|infinite]]. It turns out the rational numbers can actually be placed in one-to-one correspondence with the integers, and therefore the set of rational numbers is the same size (''cardinality'') as the set of integers: they are both [[countable set]]s.
| |
| | |
| Cantor gave two proofs that the cardinality of the set of [[integer]]s is strictly smaller than that of the set of [[real number]]s (see [[Cantor's first uncountability proof]] and [[Cantor's diagonal argument]]). His proofs, however, give no indication of the extent to which the cardinality of the integers is less than that of the real numbers. Cantor proposed the continuum hypothesis as a possible solution to this question.
| |
| | |
| The hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. Equivalently, as the [[Cardinal number|cardinality]] of the integers is <math>\aleph_0</math> ("[[aleph number#Aleph-naught|aleph-naught]]") and the [[cardinality of the real numbers]] is <math>2^{\aleph_0}</math>, the continuum hypothesis says that there is no set <math>S</math> for which
| |
| | |
| :<math> \aleph_0 < |S| < 2^{\aleph_0}. \,</math>
| |
| | |
| Assuming the [[axiom of choice]], [[aleph numbers|there is a smallest cardinal number <math>\aleph_1</math> greater than <math>\aleph_0</math>]], and the continuum hypothesis is in turn equivalent to the equality
| |
| | |
| :<math>2^{\aleph_0} = \aleph_1. \, </math>
| |
| | |
| There is also a generalization of the continuum hypothesis called the '''generalized continuum hypothesis''' ('''GCH''') which says that for all [[Ordinal number|ordinals]] <math>\alpha\,</math>
| |
| | |
| :<math>2^{\aleph_\alpha} = \aleph_{\alpha+1}.</math>
| |
| | |
| A consequence of the continuum hypothesis is that every infinite [[subset]] of the real numbers either has the same cardinality as the integers or the same cardinality as the entire set of the reals.
| |
| | |
| ==Impossibility of proof and disproof in ZFC==
| |
| Cantor believed the continuum hypothesis to be true and tried for many years to [[mathematical proof|prove]] it, in vain. It became the first on David Hilbert's [[Hilbert's problems|list of important open questions]] that was presented at the [[International Congress of Mathematicians]] in the year 1900 in Paris. [[Axiomatic set theory]] was at that point not yet formulated.
| |
| | |
| [[Kurt Gödel]] showed in 1940<!-- earlier, deleted paragraph said 1939; which is right? --> that the continuum hypothesis (CH for short) cannot be disproved from the standard [[Zermelo–Fraenkel set theory]] (ZF), even if the [[axiom of choice]] is adopted (ZFC). [[Paul Cohen (mathematician)|Paul Cohen]] showed in 1963 that CH cannot be proven from those same axioms either. Hence, CH is ''[[independence (mathematical logic)|independent]]'' of [[ZFC]]. Both of these results assume that the Zermelo–Fraenkel axioms themselves do not contain a contradiction; this assumption is widely believed to be true.
| |
| | |
| The continuum hypothesis was not the first statement shown to be independent of ZFC. An immediate consequence of [[Gödel's incompleteness theorem]], which was published in 1931, is that there is a formal statement (one for each appropriate [[Gödel numbering]] scheme) expressing the consistency of ZFC that is independent of ZFC. The continuum hypothesis and the [[axiom of choice]] were among the first mathematical statements shown to be independent of ZF set theory. These independence proofs were not completed until Paul Cohen developed [[Forcing (mathematics)|forcing]] in the 1960s.
| |
| | |
| The continuum hypothesis is closely related to many statements in [[mathematical analysis|analysis]], point set [[topology]] and [[measure theory]]. As a result of its independence, many substantial [[conjecture]]s in those fields have subsequently been shown to be independent as well.
| |
| | |
| So far, CH appears to be independent of all known ''[[large cardinal axiom]]s'' in the context of ZFC.
| |
| | |
| Gödel and Cohen's negative results are not universally accepted as disposing of the hypothesis, and Hilbert's problem remains an active topic of contemporary research (see Woodin 2001a). Koellner (2011a) has also written an overview of the status of current research into CH.
| |
| | |
| ==Arguments for and against CH==
| |
| Gödel believed that CH is false and that his proof that CH is consistent with ZFC only shows that the [[Zermelo–Fraenkel set theory|Zermelo–Fraenkel]] axioms do not adequately characterize the universe of sets. Gödel was a [[Philosophy of mathematics#Platonism|platonist]] and therefore had no problems with asserting the truth and falsehood of statements independent of their provability. Cohen, though a [[Formalism (mathematics)|formalist]] {{harv|Goodman|1979}}, also tended towards rejecting CH.
| |
| | |
| Historically, mathematicians who favored a "rich" and "large" [[universe (mathematics)|universe]] of sets were against CH, while those favoring a "neat" and "controllable" universe favored CH. Parallel arguments were made for and against the [[axiom of constructibility]], which implies CH. More recently, [[Matthew Foreman]] has pointed out that [[ontological maximalism]] can actually be used to argue in favor of CH, because among models that have the same reals, models with "more" sets of reals have a better chance of satisfying CH (Maddy 1988, p. 500).
| |
| | |
| Another viewpoint is that the conception of set is not specific enough to determine whether CH is true or false. This viewpoint was advanced as early as 1923 by [[Skolem]], even before Gödel's first incompleteness theorem. Skolem argued on the basis of what is now known as [[Skolem's paradox]], and it was later supported by the independence of CH from the axioms of ZFC, since these axioms are enough to establish the elementary properties of sets and cardinalities. In order to argue against this viewpoint, it would be sufficient to demonstrate new axioms that are supported by intuition and resolve CH in one direction or another. Although the [[axiom of constructibility]] does resolve CH, it is not generally considered to be intuitively true any more than CH is generally considered to be false (Kunen 1980, p. 171).
| |
| | |
| At least two other axioms have been proposed that have implications for the continuum hypothesis, although these axioms have not currently found wide acceptance in the mathematical community. In 1986, Chris Freiling presented an argument against CH by showing that the negation of CH is equivalent to [[Freiling's axiom of symmetry]], a statement about [[probability|probabilities]]. Freiling believes this axiom is "intuitively true" but others have disagreed. A difficult argument against CH developed by [[W. Hugh Woodin]] has attracted considerable attention since the year 2000 (Woodin 2001a, 2001b). Foreman (2003) does not reject Woodin's argument outright but urges caution.
| |
| | |
| [[Solomon Feferman]] (2011) has made a complex philosophical argument that CH is not a definite mathematical problem. He proposes a theory of "definiteness" using a semi-intuitionistic subsystem of ZF that accepts [[classical logic]] for bounded quantifiers but uses [[intuitionistic logic]] for unbounded ones, and suggests that a proposition <math>\phi</math> is mathematically "definite" if the semi-intuitionistic theory can prove <math>(\phi \or \neg\phi)</math>. He conjectures that CH is not definite according to this notion, and proposes that CH should therefore be considered not to have a truth value. [[Peter Koellner]] (2011b) wrote a critical commentary on Feferman's article.
| |
| | |
| [[Joel David Hamkins]] proposes a multiverse approach to set theory and argues that "the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for." (Hamkins 2012).
| |
| | |
| ==The generalized continuum hypothesis ==<!-- This section is linked from [[Forcing (mathematics)]] -->
| |
| The ''generalized continuum hypothesis'' (GCH) states that if an infinite set's cardinality lies between that of an infinite set ''S'' and that of the [[power set]] of ''S'', then it either has the same cardinality as the set ''S'' or the same cardinality as the power set of ''S''. That is, for any [[infinite set|infinite]] [[cardinal number|cardinal]] <math>\lambda\,</math> there is no cardinal <math>\kappa\,</math> such that <math>\lambda <\kappa <2^{\lambda}.\,</math> GCH is equivalent to:
| |
| :<math>\aleph_{\alpha+1}=2^{\aleph_\alpha}</math> for every [[ordinal number|ordinal]] <math>\alpha.\,</math> | |
| The [[beth number]]s provide an alternate notation for this condition: <math>\aleph_\alpha=\beth_\alpha</math> for every ordinal <math>\alpha.\,</math>
| |
| | |
| This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the [[power set]] of the integers.
| |
| | |
| Like CH, GCH is also independent of ZFC, but [[Wacław Sierpiński|Sierpiński]] proved that ZF + GCH implies the [[axiom of choice]] (AC), so choice and GCH are not independent in ZF; there are no models of ZF in which GCH holds and AC fails. To prove this, Sierpiński showed GCH implies that every cardinality n is smaller than some [[Aleph number]], and thus can be ordered. This is done by showing that n is smaller than <math>2^{\aleph_0+n}\,</math> which is smaller than its own [[Hartogs number]] (this uses the equality <math>2^{\aleph_0+n}\, = \,2\cdot\,2^{\aleph_0+n} </math>; for the full proof, see Gillman (2002).
| |
| | |
| [[Kurt Gödel]] showed that GCH is a consequence of ZF + [[Axiom of constructibility|V=L]] (the axiom that every set is constructible relative to the ordinals), and is therefore consistent with ZFC. As GCH implies CH, Cohen's model in which CH fails is a model in which GCH fails, and thus GCH is not provable from ZFC. W. B. Easton used the method of forcing developed by Cohen to prove [[Easton's theorem]], which shows it is consistent with ZFC for arbitrarily large cardinals <math>\aleph_\alpha</math> to fail to satisfy <math>2^{\aleph_\alpha} = \aleph_{\alpha + 1}.</math> Much later, [[Matthew Foreman|Foreman]] and [[W. Hugh Woodin|Woodin]] proved that (assuming the consistency of very large cardinals) it is consistent that <math>2^\kappa>\kappa^+\,</math> holds for every infinite cardinal <math>\kappa.\,</math> Later Woodin extended this by showing the consistency of <math>2^\kappa=\kappa^{++}\,</math> for every <math>\kappa\,</math>. A recent result of Carmi Merimovich shows that, for each ''n''≥1, it is consistent with ZFC that for each κ, 2<sup>κ</sup> is the ''n''th successor of κ. On the other hand, {{harvs|first=László |last=Patai|year=1930|txt}} proved, that if γ is an ordinal and for each infinite cardinal κ, 2<sup>κ</sup> is the γth successor of κ, then γ is finite.
| |
| | |
| For any infinite sets A and B, if there is an injection from A to B then there is an injection from subsets of A to subsets of B. Thus for any infinite cardinals A and B,
| |
| :<math>A < B \to 2^A \le 2^B.</math>
| |
| If A and B are finite, the stronger inequality
| |
| :<math>A < B \to 2^A < 2^B \!</math>
| |
| holds. GCH implies that this strict, stronger inequality holds for infinite cardinals as well as finite cardinals.
| |
| | |
| ===Implications of GCH for cardinal exponentiation===
| |
| Although the Generalized Continuum Hypothesis refers directly only to cardinal exponentiation with 2 as the base, one can deduce from it the values of cardinal exponentiation in all cases. It implies that <math>\aleph_{\alpha}^{\aleph_{\beta}}</math> is (see: Hayden & Kennison (1968), page 147, exercise 76):
| |
| :<math>\aleph_{\beta+1}</math> when α ≤ β+1; | |
| :<math>\aleph_{\alpha}</math> when β+1 < α and <math>\aleph_{\beta} < \operatorname{cf} (\aleph_{\alpha})</math> where '''cf''' is the [[cofinality]] operation; and
| |
| :<math>\aleph_{\alpha+1}</math> when β+1 < α and <math>\aleph_{\beta} \ge \operatorname{cf} (\aleph_{\alpha})</math>.
| |
| | |
| ==See also==
| |
| *[[Aleph number]]
| |
| *[[Beth number]]
| |
| *[[Cardinality]]
| |
| *[[Ω-logic]]
| |
| | |
| ==References==
| |
| <!--
| |
| <references/>
| |
| | |
| This article uses parenthetical citations. Please do not add <references/>, {{reflist}}, etc. - instead, fix new citations so that they use the existing citation style
| |
| -->
| |
| *{{Cite book
| |
| | first = P. J. | last = Cohen
| |
| | title = Set Theory and the Continuum Hypothesis
| |
| | publisher = W. A. Benjamin
| |
| | year = 1966
| |
| }}
| |
| *{{Cite journal
| |
| | first = Paul J. | last = Cohen
| |
| | title = The Independence of the Continuum Hypothesis
| |
| | journal = Proceedings of the National Academy of Sciences of the United States of America
| |
| | volume = 50 | issue = 6 | date = December 15, 1963 | pages = 1143–1148
| |
| | doi = 10.1073/pnas.50.6.1143
| |
| | pmid = 16578557
| |
| | pmc = 221287 | jstor=71858
| |
| }}
| |
| *{{Cite journal
| |
| | first = Paul J. | last = Cohen
| |
| | title = The Independence of the Continuum Hypothesis, II
| |
| | journal = Proceedings of the National Academy of Sciences of the United States of America
| |
| | volume = 51 | issue = 1 | date = January 15, 1964 | pages = 105–110
| |
| | doi = 10.1073/pnas.51.1.105
| |
| | pmid = 16591132
| |
| | pmc = 300611 | jstor=72252
| |
| }}
| |
| *{{Cite book
| |
| | first = H. G. | last = Dales
| |
| | coauthors = W. H. Woodin
| |
| | title = An Introduction to Independence for Analysts
| |
| | publisher = Cambridge
| |
| | year = 1987
| |
| }}
| |
| *{{Cite book
| |
| | first = Herbert | last = Enderton
| |
| | year = 1977
| |
| | title = Elements of Set Theory
| |
| | publisher = Academic Press
| |
| }}
| |
| *{{Cite web
| |
| | author = Feferman, Solomon
| |
| | year = 2011
| |
| | authorlink = Solomon Feferman
| |
| | url = http://math.stanford.edu/~feferman/papers/IsCHdefinite.pdf
| |
| | title = Is the Continuum Hypothesis a definite mathematical problem?
| |
| | work = [http://logic.harvard.edu/efi.php Exploring the Frontiers of Independence] (Harvard lecture series)
| |
| }} ([http://math.stanford.edu/~feferman/papers/IsCHdefinite-slides.pdf lecture slides])
| |
| *{{Cite web
| |
| | author = Foreman, Matt
| |
| | year = 2003
| |
| | url = http://www.math.helsinki.fi/logic/LC2003/presentations/foreman.pdf
| |
| | title = Has the Continuum Hypothesis been Settled?
| |
| | format = PDF
| |
| | accessdate = February 25, 2006
| |
| }}
| |
| *{{Cite journal
| |
| | first = Chris | last = Freiling
| |
| | title = Axioms of Symmetry: Throwing Darts at the Real Number Line
| |
| | journal = Journal of Symbolic Logic
| |
| | volume = 51 | issue = 1 | year = 1986 | pages = 190–200
| |
| | doi = 10.2307/2273955
| |
| | publisher = Association for Symbolic Logic
| |
| | jstor=2273955}}
| |
| *{{Cite book
| |
| | first = K. | last = Gödel
| |
| | title = The Consistency of the Continuum-Hypothesis
| |
| | publisher = Princeton University Press
| |
| | year = 1940
| |
| }}
| |
| * {{Cite journal
| |
| | first = Leonard
| |
| | last = Gillman
| |
| | title = Two Classical Surprises Concerning the Axiom of Choice and the Continuum Hypothesis
| |
| | journal = American Mathematical Monthly
| |
| | year = 2002
| |
| | volume = 109
| |
| | url = http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gillman544-553.pdf
| |
| }}
| |
| *Gödel, K.: ''What is Cantor's Continuum Problem?'', reprinted in Benacerraf and Putnam's collection ''Philosophy of Mathematics'', 2nd ed., Cambridge University Press, 1983. An outline of Gödel's arguments against CH.
| |
| *{{cite journal
| |
| | last = Goodman | first = Nicolas D.
| |
| | doi = 10.2307/2320581
| |
| | issue = 7
| |
| | journal = The American Mathematical Monthly
| |
| | mr = 542765
| |
| | pages = 540–551
| |
| | quote = This view is often called formalism. Positions more or less like this may be found in Haskell Curry [5], Abraham Robinson [17], and Paul Cohen [4].
| |
| | title = Mathematics as an objective science
| |
| | volume = 86
| |
| | year = 1979
| |
| | ref=harv}}
| |
| *Hamkins, Joel David. The set-theoretic multiverse. Rev. Symb. Log. 5 (2012), no. 3, 416–449.
| |
| *[[Seymour Hayden]] and [[John F. Kennison]]: ''Zermelo–Fraenkel Set Theory'' (1968), Charles E. Merrill Publishing Company, Columbus, Ohio.
| |
| *{{cite web
| |
| | first = Peter | last = Koellner
| |
| | authorlink = Peter Koellner
| |
| | url = http://logic.harvard.edu/EFI_CH.pdf
| |
| | title = The Continuum Hypothesis
| |
| | work = Exploring the Frontiers of Independence (Harvard lecture series)
| |
| | year = 2011a
| |
| }}
| |
| *{{cite web
| |
| | first = Peter | last = Koellner
| |
| | url = http://logic.harvard.edu/EFI_Feferman_comments.pdf
| |
| | title = Feferman On the Indefiniteness of CH
| |
| | year = 2011b <!-- not sure of this, it might be 2012 -->
| |
| }}
| |
| *{{Cite book
| |
| | last1=Kunen
| |
| | first1=Kenneth
| |
| | author1-link=Kenneth Kunen
| |
| | title=[[Set Theory: An Introduction to Independence Proofs]]
| |
| | publisher=North-Holland
| |
| | location=Amsterdam
| |
| | isbn=978-0-444-85401-8
| |
| | year=1980
| |
| }}
| |
| *{{Cite journal
| |
| | first = Penelope | last = Maddy
| |
| | journal = Journal of Symbolic Logic
| |
| | title = Believing the Axioms, I
| |
| | volume = 53 | issue = 2 |date=June 1988 | pages = 481–511
| |
| | doi = 10.2307/2274520
| |
| | publisher = Association for Symbolic Logic
| |
| | jstor=2274520}}
| |
| * Martin, D. (1976). "Hilbert's first problem: the continuum hypothesis," in ''Mathematical Developments Arising from Hilbert's Problems,'' Proceedings of Symposia in Pure Mathematics XXVIII, F. Browder, editor. American Mathematical Society, 1976, pp. 81–92. ISBN 0-8218-1428-1
| |
| *{{Cite web
| |
| | author = McGough, Nancy
| |
| | url = http://www.ii.com/math/ch/
| |
| | title = The Continuum Hypothesis
| |
| }}
| |
| *{{Cite journal
| |
| | first = Carmi | last = Merimovich
| |
| | title = A power function with a fixed finite gap everywhere
| |
| | journal = Journal of Symbolic Logic
| |
| | volume = 72 | issue = 2 | year = 2007 | pages = 361–417
| |
| | doi = 10.2178/jsl/1185803615
| |
| }}
| |
| *{{cite journal
| |
| | last = Patai | first = L.
| |
| | journal = Mathematische und naturwissenschaftliche Berichte aus Ungarn
| |
| | pages = 127–142
| |
| | title = Untersuchungen über die א-reihe
| |
| | volume = 37
| |
| | year = 1930
| |
| | ref = harv}}
| |
| *{{Cite journal
| |
| | first = W. Hugh | last = Woodin
| |
| | title = The Continuum Hypothesis, Part I
| |
| | journal = Notices of the AMS
| |
| | volume = 48 | issue = 6 | year = 2001a | pages = 567–576
| |
| | url = http://www.ams.org/notices/200106/fea-woodin.pdf
| |
| |format=PDF}}
| |
| *{{Cite journal
| |
| | first = W. Hugh | last = Woodin
| |
| | title = The Continuum Hypothesis, Part II
| |
| | journal = Notices of the AMS
| |
| | volume = 48 | issue = 7 | year = 2001b | pages = 681–690
| |
| | url = http://www.ams.org/notices/200107/fea-woodin.pdf
| |
| |format=PDF}}
| |
| | |
| ; Primary literature in German:
| |
| * {{Citation |surname=Cantor|given=Georg|year=1878|url=http://www.digizeitschriften.de/dms/img/?PPN=PPN243919689_0084&DMDID=dmdlog15|title=Ein Beitrag zur Mannigfaltigkeitslehre|journal=[[Journal für die Reine und Angewandte Mathematik]]|volume=84|pages=242–258}}.
| |
| | |
| == External links ==
| |
| *{{MathWorld |title=Continuum Hypothesis |id=ContinuumHypothesis |author=[[Matthew Szudzik|Szudzik, Matthew]] and [[Eric W. Weisstein|Weisstein, Eric W.]] }}
| |
| | |
| {{PlanetMath attribution|id=1184|title=Generalized continuum hypothesis}}
| |
| {{Set theory}}
| |
| {{Hilbert's problems}}
| |
| | |
| {{DEFAULTSORT:Continuum Hypothesis}}
| |
| [[Category:Forcing (mathematics)]]
| |
| [[Category:Independence results]]
| |
| [[Category:Basic concepts in infinite set theory]]
| |
| [[Category:Hilbert's problems]]
| |
| [[Category:Infinity]]
| |
| [[Category:Hypotheses]]
| |
| [[Category:Cardinal numbers]]
| |
| {{Link FA|lmo}}
| |