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{{Short description|Proposition in mathematical logic}} | |||
{{About|the hypothesis in set theory|the assumption in fluid mechanics|Fluid mechanics}} | |||
{{About|the hypothesis in set theory|the assumption in fluid mechanics|Continuum assumption|the album by Epoch of Unlight|The Continuum Hypothesis (album)}} | |||
In ], the '''continuum hypothesis''' (abbreviated '''CH''') is a ], advanced by ] in 1878, about the possible sizes of ] ]. It states: | |||
{{Use shortened footnotes|date=May 2021}} | |||
:''There is no set whose cardinality is strictly between that of the integers and that of the real numbers.'' | |||
Establishing the truth or falsehood of the continuum hypothesis is the first of ] presented in the year 1900. The contributions of ] in 1940 and ] in 1963 showed that the hypothesis can neither be disproved nor be ] using the axioms of ], the standard foundation of modern mathematics, provided ZF set theory is ]. | |||
In ], specifically ], the '''continuum hypothesis''' (abbreviated '''CH''') is a hypothesis about the possible sizes of ]s. It states: | |||
The name of the hypothesis comes from the term ] for the real numbers. | |||
{{Blockquote|There is no set whose ] is strictly between that of the ]s and the ]s.}} | |||
Or equivalently: | |||
{{Blockquote|Any subset of the real numbers is either finite, or countably infinite, or has the cardinality of the real numbers.}} | |||
In ] with the ] (ZFC), this is equivalent to the following equation in ]s: <math>2^{\aleph_0}=\aleph_1</math>, or even shorter with ]s: <math>\beth_1 = \aleph_1</math>. | |||
The continuum hypothesis was advanced by ] in 1878,{{r|Cantor1878}} and establishing its truth or falsehood is the first of ] presented in 1900. The answer to this problem is ] of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by ], complementing earlier work by ] in 1940.{{r|Gödel1940}} | |||
The name of the hypothesis comes from the term '']'' for the real numbers. | |||
==History== | |||
Cantor believed the continuum hypothesis to be true and for many years tried in vain to prove it.{{r|Dauben1990_1347}} It became the first on David Hilbert's ] that was presented at the ] in the year 1900 in Paris. ] was at that point not yet formulated. | |||
] proved in 1940 that the negation of the continuum hypothesis, i.e., the existence of a set with intermediate cardinality, could not be proved in standard set theory.{{r|Gödel1940}} The second half of the independence of the continuum hypothesis – i.e., unprovability of the nonexistence of an intermediate-sized set – was proved in 1963 by ].{{r|Cohen1963}} | |||
==Cardinality of infinite sets== | ==Cardinality of infinite sets== | ||
{{Main|Cardinal number}} | {{Main|Cardinal number}} | ||
Two sets are said to have the same '']'' or '']'' if there exists a ] (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}. | Two sets are said to have the same '']'' or '']'' if there exists a ] (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 ]s or ]s, |
With infinite sets such as the set of ]s or ]s, the existence of a bijection between two sets becomes more difficult 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 is flawed; it does not take proper account of the fact that all three sets are ]. 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 ]s. | ||
Cantor gave two proofs that the cardinality of the set of ]s is strictly smaller than that of the set of ]s (see ] and ]). 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. | Cantor gave two proofs that the cardinality of the set of ]s is strictly smaller than that of the set of ]s (see ] and ]). 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. |
The continuum hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. That is, every set, ''S'', of real numbers can either be mapped one-to-one into the integers or the real numbers can be mapped one-to-one into ''S''. As the real numbers are ] with the ] of the integers, i.e. <math>|\mathbb{R}|=2^{\aleph_0}</math>, the continuum hypothesis can be restated as follows: | ||
{{math theorem | |||
:<math> \aleph_0 < |S| < 2^{\aleph_0}. \,</math> | |||
| <math>\nexists S\colon\aleph_0 < |S| < 2^{\aleph_0}</math>. | |||
|name=Continuum hypothesis}} | |||
Assuming the ], |
Assuming the ], there is a unique 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>.{{r|Goldrei1996}} | ||
==Independence from ZFC== | |||
:<math>2^{\aleph_0} = \aleph_1. \, </math> | |||
The independence of the continuum hypothesis (CH) from ] (ZF) follows from combined work of ] and ]. | |||
Gödel<ref>{{cite journal|doi=10.1073/pnas.24.12.556 |title=The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis |date=1938 |last1=Gödel |first1=Kurt |journal=Proceedings of the National Academy of Sciences |volume=24 |issue=12 |pages=556–557 |pmid=16577857 |pmc=1077160 |bibcode=1938PNAS...24..556G |doi-access=free }}</ref>{{r|Gödel1940}} showed that CH cannot be disproved from ZF, even if the ] (AC) is adopted (making ZFC). Gödel's proof shows that CH and AC both hold in the ] L, an ] of ZF set theory, assuming only the axioms of ZF. The existence of an inner model of ZF in which additional axioms hold shows that the additional axioms are ] with ZF, provided ZF itself is consistent. The latter condition cannot be proved in ZF itself, due to ], but is widely believed to be true and can be proved in stronger set theories. | |||
There is also a generalization of the continuum hypothesis called the '''generalized continuum hypothesis''' ('''GCH''') which says that for all ] <math>\alpha\,</math> | |||
Cohen{{r|Cohen1963|Cohen1964}} showed that CH cannot be proven from the ZFC axioms, completing the overall independence proof. To prove his result, Cohen developed the method of ], which has become a standard tool in set theory. Essentially, this method begins with a model of ZF in which CH holds, and constructs another model which contains more sets than the original, in a way that CH does not hold in the new model. Cohen was awarded the ] in 1966 for his proof. | |||
:<math>2^{\aleph_\alpha} = \aleph_{\alpha+1}.</math> | |||
The independence proof just described shows that CH is independent of ZFC. Further research has shown that CH is independent of all known '']s'' in the context of ZFC.{{r|Feferman1999_99111}} Moreover, it has been shown that the ] can be any cardinal consistent with ]. A result of Solovay, proved shortly after Cohen's result on the independence of the continuum hypothesis, shows that in any model of ZFC, if <math>\kappa</math> is a cardinal of uncountable ], then there is a forcing extension in which <math>2^{\aleph_0} = \kappa</math>. However, per König's theorem, it is not consistent to assume <math>2^{\aleph_0}</math> is <math>\aleph_\omega</math> or <math>\aleph_{\omega_1+\omega}</math> or any cardinal with cofinality <math>\omega</math>. | |||
A consequence of the hypothesis is that every infinite ] of the real numbers either has the same cardinality as the integers or the same cardinality as the entire set of the reals. | |||
The continuum hypothesis is closely related to many statements in ], point set ] and ]. As a result of its independence, many substantial ]s in those fields have subsequently been shown to be independent as well. | |||
==Impossibility of proof and disproof in ZFC== | |||
Cantor believed the continuum hypothesis to be true and tried for many years to ] it, in vain. It became the first on David Hilbert's ] that was presented at the ] in the year 1900 in Paris. ] was at that point not yet formulated. | |||
The independence from ZFC means that proving or disproving the CH within ZFC is impossible. However, Gödel and Cohen's negative results are not universally accepted as disposing of all interest in the continuum hypothesis. The continuum hypothesis remains an active topic of research; see ]{{r|Woodin2001a|Woodin2001b}} and ]{{r|Koellner2011a}} for an overview of the current research status. | |||
] showed in 1940<!-- earlier, deleted paragraph said 1939; which is right? --> that the continuum hypothesis (CH for short) cannot be disproved from the standard ] (ZF), even if the ] is adopted (ZFC). ] showed in 1963 that CH cannot be proven from those same axioms either. Hence, CH is '']'' of ]. 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 |
The continuum hypothesis and the ] were among the first genuinely mathematical statements shown to be independent of ZF set theory. Although the existence of some statements independent of ZFC had already been known more than two decades prior: for example, assuming ] and the consistency ZFC, ], which were published in 1931, establish that there is a formal statement (one for each appropriate ] scheme) expressing the consistency of ZFC, that is also independent of it. The latter independence result indeed holds for many theories. | ||
==Arguments for and against the continuum hypothesis== | |||
The continuum hypothesis is closely related to many statements in ], point set ] and ]. As a result of its independence, many substantial ]s in those fields have subsequently been shown to be independent as well. | |||
Gödel believed that CH is false, and that his proof that CH is consistent with ZFC only shows that the ] axioms do not adequately characterize the universe of sets. Gödel was a ] and therefore had no problems with asserting the truth and falsehood of statements independent of their provability. Cohen, though a ],{{r|Goodman1979}} also tended towards rejecting CH. | |||
Historically, mathematicians who favored a "rich" and "large" ] of sets were against CH, while those favoring a "neat" and "controllable" universe favored CH. Parallel arguments were made for and against the ], which implies CH. More recently, ] has pointed out that ] 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.{{sfn|Maddy|1988|p=500}} | |||
So far, CH appears to be independent of all known '']s'' in the context of ZFC. | |||
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 ], even before Gödel's first incompleteness theorem. Skolem argued on the basis of what is now known as ], 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 ] does resolve CH, it is not generally considered to be intuitively true any more than CH is generally considered to be false.{{r|Kunen1980_171}} | |||
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. | |||
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{{r|Freiling1986}} presented an argument against CH by showing that the negation of CH is equivalent to ], a statement derived by arguing from particular intuitions about ]. Freiling believes this axiom is "intuitively clear"{{r|Freiling1986}} but others have disagreed.{{r|bagemihl|Hamkins2015}} | |||
==Arguments for and against CH== | |||
Gödel believed that CH is false and that his proof that CH is ] only shows that the ] axioms do not adequately describe the universe of sets. Gödel was a ] and therefore had no problems with asserting the truth and falsehood of statements independent of their provability. Cohen, though a ]{{citation needed|date=November 2011}}, also tended towards rejecting CH. | |||
A difficult argument against CH developed by ] has attracted considerable attention since the year 2000.{{r|Woodin2001a|Woodin2001b}} ] does not reject Woodin's argument outright but urges caution.{{r|Foreman2003}} Woodin proposed a new hypothesis that he labeled the {{nowrap|"(*)-axiom"}}, or "Star axiom". The Star axiom would imply that <math>2^{\aleph_0}</math> is <math>\aleph_2</math>, thus falsifying CH. The Star axiom was bolstered by an independent May 2021 proof showing the Star axiom can be derived from a variation of ]. However, Woodin stated in the 2010s that he now instead believes CH to be true, based on his belief in his new "ultimate L" conjecture.<ref name="quanta 2021">{{cite news |last1=Wolchover |first1=Natalie |title=How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer. |url=https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/ |access-date=30 December 2021 |work=Quanta Magazine |date=15 July 2021 |language=en}}</ref><ref>{{cite journal |last1=Rittberg |first1=Colin J. |title=How Woodin changed his mind: new thoughts on the Continuum Hypothesis |journal=Archive for History of Exact Sciences |date=March 2015 |volume=69 |issue=2 |pages=125–151 |doi=10.1007/s00407-014-0142-8|s2cid=122205863 }}</ref> | |||
Historically, mathematicians who favored a "rich" and "large" ] of sets were against CH, while those favoring a "neat" and "controllable" universe favored CH. Parallel arguments were made for and against the ], which implies CH. More recently, ] has pointed out that ] 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). | |||
] argued that CH is not a definite mathematical problem.{{r|Feferman2011}} He proposed a theory of "definiteness" using a semi-intuitionistic subsystem of ZF that accepts ] for bounded quantifiers but uses ] for unbounded ones, and suggested that a proposition <math>\phi</math> is mathematically "definite" if the semi-intuitionistic theory can prove <math>(\phi \lor \neg\phi)</math>. He conjectured that CH is not definite according to this notion, and proposed that CH should, therefore, be considered not to have a truth value. ] wrote a critical commentary on Feferman's article.{{r|Koellner2011b}} | |||
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 ], even before Gödel's first incompleteness theorem. Skolem argued on the basis of what is now known as ], 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 ] 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). | |||
] proposes a ] 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".{{r|Hamkins2012}} In a related vein, ] wrote that he does "not agree with the pure Platonic view that the interesting problems in set theory can be decided, that we just have to discover the additional axiom. My mental picture is that we have many possible set theories, all conforming to ZFC".{{r|Shelah2003}} | |||
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 ], a statement about ]. Freiling believes this axiom is "intuitively true" but others have disagreed. A difficult argument against CH developed by ] has attracted considerable attention since the year 2000 (Woodin 2001a, 2001b). Foreman (2003) does not reject Woodin's argument outright but urges caution. | |||
==Generalized continuum hypothesis==<!-- This section is linked from ] --> | |||
] (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 ] for bounded quantifiers but uses ] 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 to not have a truth value. Koellner (2011b) wrote a critical commentary on Feferman's article. | |||
The ''generalized continuum hypothesis'' (GCH) states that if an infinite set's cardinality lies between that of an infinite set {{mvar|S}} and that of the ] <math>\mathcal{P}(S)</math> of {{mvar|S}}, then it has the same cardinality as either {{mvar|S}} or <math>\mathcal{P}(S)</math>. That is, for any ] cardinal <math>\lambda</math> there is no cardinal <math>\kappa</math> such that <math>\lambda <\kappa <2^{\lambda}</math>. GCH is equivalent to: | |||
{{block indent|<math>\aleph_{\alpha+1}=2^{\aleph_\alpha}</math> for every ] <math>\alpha</math>{{r|Goldrei1996}}}} | |||
==The generalized continuum hypothesis ==<!-- This section is linked from ] --> | |||
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 ] 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 ] ] <math>\lambda\,</math> there is no cardinal <math>\kappa\,</math> such that <math>\lambda <\kappa <2^{\lambda}.\,</math> An equivalent condition is that <math>\aleph_{\alpha+1}=2^{\aleph_\alpha}</math> for every ] <math>\alpha.\,</math> The ]s provide an alternate notation for this condition: <math>\aleph_\alpha=\beth_\alpha</math> for every ordinal <math>\alpha.\,</math> | |||
(occasionally called ''Cantor's aleph hypothesis''). | |||
This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the ] of the integers. Like CH, GCH is also independent of ZFC, but ] proved that ZF + GCH implies the ] (AC), so choice and GCH are not independent in ZF; there are no models of ZF in which GCH holds and AC fails. | |||
The ]s provide an alternative notation for this condition: <math>\aleph_\alpha=\beth_\alpha</math> for every ordinal <math>\alpha</math>. The continuum hypothesis is the special case for the ordinal <math>\alpha=1</math>. GCH was first suggested by ].{{r|Jourdain1905}} For the early history of GCH, see Moore.{{r|Moore2011}} | |||
] showed that GCH is a consequence of ZF + ] (the axiom that every set is constructible relative to the ordinals), and is 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 ], 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, ] and ] 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, Laszlo Patai 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. {{Citation needed|date=September 2010}} | |||
Like CH, GCH is also independent of ZFC, but ] proved that ZF + GCH implies the ] (AC) (and therefore the negation of the ], AD), 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 ], 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 ]—this uses the equality <math>2^{\aleph_0+n}\, = \,2\cdot\,2^{\aleph_0+n} </math>; for the full proof, see Gillman.{{r|Gillman2002}} | |||
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>. | |||
] showed that GCH is a consequence of ZF + ] (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 ], 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, ] and ] 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 {{nowrap|<math>\kappa</math>.}} Carmi Merimovich{{r|Merimovich2007}} showed that, for each {{math|''n'' ≥ 1}}, it is consistent with ZFC that for each infinite cardinal {{mvar|κ}}, {{math|2<sup>''κ''</sup>}} is the {{mvar|n}}th successor of {{mvar|κ}} (assuming the consistency of some large cardinal axioms). On the other hand, László Patai{{r|Patai1930}} proved that if {{mvar|γ}} is an ordinal and for each infinite cardinal {{mvar|κ}}, {{math|2<sup>''κ''</sup>}} is the {{mvar|γ}}th successor of {{mvar|κ}}, then {{mvar|γ}} is finite. | |||
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. | For any infinite sets {{mvar|A}} and {{mvar|B}}, if there is an injection from {{mvar|A}} to {{mvar|B}} then there is an injection from subsets of {{mvar|A}} to subsets of {{mvar|B}}. Thus for any infinite cardinals {{mvar|A}} and {{mvar|B}}, <math>A < B \to 2^A \le 2^B</math>. If {{mvar|A}} and {{mvar|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=== | ===Implications of GCH for cardinal exponentiation=== | ||
Although the |
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 <math>\aleph_{\alpha}^{\aleph_{\beta}}</math> in all cases. GCH implies that for ordinals {{mvar|α}} and {{mvar|β}}:{{r|HaydenKennison1968}} | ||
:<math>\aleph_{\beta+1}</math> when α ≤ β+1; | |||
*<math>\aleph_{\alpha}^{\aleph_{\beta}} = \aleph_{\beta+1}</math> when {{math|''α'' ≤ ''β''+1}}; | |||
*<math>\aleph_{\alpha}^{\aleph_{\beta}} = \aleph_{\alpha}</math> when {{math|''β''+1 < ''α''}} and <math>\aleph_{\beta} < \operatorname{cf} (\aleph_{\alpha})</math>, where '''cf''' is the ] operation; and | |||
*<math>\aleph_{\alpha}^{\aleph_{\beta}} = \aleph_{\alpha+1}</math> when {{math|''β''+1 < ''α''}} and {{nowrap|<math>\aleph_{\beta} \ge \operatorname{cf} (\aleph_{\alpha})</math>.}} | |||
The first equality (when {{mvar|''α'' ≤ ''β''+1}}) follows from: | |||
<math display="block">\aleph_{\alpha}^{\aleph_{\beta}} \le \aleph_{\beta+1}^{\aleph_{\beta}} =(2^{\aleph_{\beta}})^{\aleph_{\beta}} = 2^{\aleph_{\beta}\cdot\aleph_{\beta}} = 2^{\aleph_{\beta}} = \aleph_{\beta+1} </math> | |||
while: | |||
<math display="block">\aleph_{\beta+1} = 2^{\aleph_{\beta}} \le \aleph_{\alpha}^{\aleph_{\beta}} .</math> | |||
The third equality (when {{mvar|''β''+1 < ''α''}} and <math>\aleph_{\beta} \ge \operatorname{cf}(\aleph_{\alpha})</math>) follows from: | |||
<math display="block">\aleph_{\alpha}^{\aleph_{\beta}} \ge \aleph_{\alpha}^{\operatorname{cf}(\aleph_{\alpha})} > \aleph_{\alpha} </math> | |||
by ], while: | |||
<math display="block">\aleph_{\alpha}^{\aleph_{\beta}} \le \aleph_{\alpha}^{\aleph_{\alpha}} \le (2^{\aleph_{\alpha}})^{\aleph_{\alpha}} = 2^{\aleph_{\alpha}\cdot\aleph_{\alpha}} = 2^{\aleph_{\alpha}} = \aleph_{\alpha+1}</math> | |||
==See also== | ==See also== | ||
*] | *] | ||
*] | |||
*] | *] | ||
*] | *] | ||
*] | |||
*] | |||
*] | |||
==References== | ==References== | ||
<!-- The use of parenthetical citations is currently deprecated - May 2021 --> | |||
*{{Cite book | |||
{{reflist|refs= | |||
| first = P. J. | last = Cohen | |||
| title = Set Theory and the Continuum Hypothesis | |||
<ref name=bagemihl>{{cite journal | |||
| publisher = W. A. Benjamin | |||
| last = Bagemihl | first = F. | |||
| year = 1966 | |||
| issue = 1 | |||
| journal = Real Analysis Exchange | |||
| mr = 1042552 | |||
| pages = 342–345 | |||
| title = Throwing a dart at Freiling's argument against the continuum hypothesis | |||
| url = https://projecteuclid.org/journals/real-analysis-exchange/volume-15/issue-1/THROWING-A-DART-AT-FREILINGS-ARGUMENT-AGAINST-THE-CONTINUUM-HYPOTHESIS/10.2307/44152014.full | |||
| volume = 15 | |||
| year = 1989–1990| doi = 10.2307/44152014 | |||
| jstor = 44152014 | |||
}}</ref> | |||
<ref name=Cantor1878> | |||
{{Cite journal | |||
|last=Cantor |first=Georg |author-link=Georg Cantor | |||
|year=1878 | |||
|title=Ein Beitrag zur Mannigfaltigkeitslehre | |||
|journal=] | |||
|volume=1878 |issue=84 |pages=242–258 | |||
|doi=10.1515/crll.1878.84.242 | |||
|doi-broken-date=1 November 2024 |url=http://www.digizeitschriften.de/dms/img/?PPN=PPN243919689_0084&DMDID=dmdlog15 | |||
}} | |||
</ref> | |||
<ref name=Gödel1940> | |||
{{Cite book | |||
|last=Gödel |first=Kurt |author-link=Kurt Gödel | |||
|year=1940 | |||
|title=The Consistency of the Continuum-Hypothesis | |||
|publisher=Princeton University Press | |||
}} | |||
</ref> | |||
<ref name=Dauben1990_1347> | |||
{{Cite book | |||
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|year=1990 | |||
|title=Georg Cantor: His mathematics and philosophy of the infinite | |||
|pages=–137 | |||
|publisher=Princeton University Press | |||
|isbn=9780691024479 | |||
|url=https://archive.org/details/georgcantorhisma0000daub | |||
|url-access=registration | |||
}} | |||
</ref> | |||
<ref name=Cohen1963> | |||
{{Cite journal | |||
|last=Cohen |first=Paul J. | |||
|date=15 December 1963 | |||
|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 |pages=1143–1148 | |||
|doi=10.1073/pnas.50.6.1143 |pmid=16578557 | |||
|pmc=221287 |jstor=71858 |bibcode=1963PNAS...50.1143C | |||
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| title = The Independence of the Continuum Hypothesis | |||
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| volume = 50 | issue = 6 | date = December 15, 1963 | pages = 1143–1148 | |||
|year=1996 | |||
| doi = 10.1073/pnas.50.6.1143 | |||
|title=Classic Set Theory | |||
| pmid = 16578557 | |||
|publisher=] | |||
| pmc = 221287 | jstor=71858 | |||
}} | |||
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<ref name=Cohen1964> | |||
{{Cite journal | |||
|last=Cohen |first=Paul J. | |||
|date=15 January 1964 | |||
|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 |pages=105–110 | |||
|doi=10.1073/pnas.51.1.105 |pmid=16591132 | |||
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| first = Paul J. | last = Cohen | |||
<ref name=Feferman1999_99111> | |||
| title = The Independence of the Continuum Hypothesis, II | |||
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| journal = Proceedings of the National Academy of Sciences of the United States of America | |||
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| volume = 51 | issue = 1 | date = January 15, 1964 | pages = 105–110 | |||
|date=February 1999 | |||
| doi = 10.1073/pnas.51.1.105 | |||
|title=Does mathematics need new axioms? | |||
| pmid = 16591132 | |||
|journal=American Mathematical Monthly | |||
| pmc = 300611 | jstor=72252 | |||
|volume=106 |issue=2 |pages=99–111 | |||
|doi=10.2307/2589047 |jstor=2589047 | |||
|citeseerx=10.1.1.37.295 | |||
}} | |||
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<ref name=Woodin2001a> | |||
{{Cite journal | |||
|last=Woodin |first=W. Hugh | |||
|year=2001 | |||
|title=The Continuum Hypothesis, Part I | |||
|journal=Notices of the AMS | |||
|volume=48 |issue=6 |pages=567–576 | |||
|url=https://www.ams.org/notices/200106/fea-woodin.pdf |archive-url=https://ghostarchive.org/archive/20221010/https://www.ams.org/notices/200106/fea-woodin.pdf |archive-date=2022-10-10 |url-status=live | |||
}} | |||
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<ref name=Woodin2001b> | |||
{{Cite journal | |||
|last=Woodin |first=W. Hugh | |||
|year=2001 | |||
|title=The Continuum Hypothesis, Part II | |||
|journal=Notices of the AMS | |||
|volume=48 |issue=7 |pages=681–690 | |||
|url=https://www.ams.org/notices/200107/fea-woodin.pdf |archive-url=https://ghostarchive.org/archive/20221010/https://www.ams.org/notices/200107/fea-woodin.pdf |archive-date=2022-10-10 |url-status=live | |||
}} | |||
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|last=Koellner |first=Peter |author-link=Peter Koellner | |||
|year=2011 | |||
|title=The Continuum Hypothesis | |||
|work=Exploring the Frontiers of Independence | |||
|series=Harvard lecture series | |||
|url=http://logic.harvard.edu/EFI_CH.pdf |archive-url=https://web.archive.org/web/20120124183745/http://logic.harvard.edu/EFI_CH.pdf |archive-date=2012-01-24 |url-status=live | |||
}} | |||
</ref> | |||
<ref name=Koellner2011b> | |||
{{Cite web | |||
|last=Koellner |first=Peter | |||
|year=2011 | |||
|title=Feferman on the indefiniteness of CH | |||
|url=http://logic.harvard.edu/EFI_Feferman_comments.pdf |archive-url=https://web.archive.org/web/20120319061308/http://logic.harvard.edu/EFI_Feferman_comments.pdf |archive-date=2012-03-19 |url-status=live | |||
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|year=1979 | |||
|title=Mathematics as an objective science | |||
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|volume=86 |issue=7 |pages=540–551 | |||
|doi=10.2307/2320581 | |||
|jstor=2320581 | |||
|mr=542765 | |||
|quote=This view is often called ''formalism''. Positions more or less like this may be found in Haskell Curry , Abraham Robinson , and Paul Cohen . | |||
}} | |||
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|title=Axioms of Symmetry: Throwing darts at the real number line | |||
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| coauthors = W. H. Woodin | |||
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| publisher = Cambridge | |||
| |
|year=2003 | ||
|title=Has the Continuum Hypothesis been settled? | |||
|access-date=25 February 2006 | |||
|url=http://www.math.helsinki.fi/logic/LC2003/presentations/foreman.pdf |archive-url=https://ghostarchive.org/archive/20221010/http://www.math.helsinki.fi/logic/LC2003/presentations/foreman.pdf |archive-date=2022-10-10 |url-status=live | |||
}} | |||
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|year=2011 | |||
|title=Is the Continuum Hypothesis a definite mathematical problem? | |||
|work=Exploring the Frontiers of Independence | |||
|series=Harvard lecture series | |||
|url=http://math.stanford.edu/~feferman/papers/IsCHdefinite.pdf |archive-url=https://ghostarchive.org/archive/20221010/http://math.stanford.edu/~feferman/papers/IsCHdefinite.pdf |archive-date=2022-10-10 |url-status=live | |||
}} | |||
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|year=2012 | |||
|title=The set-theoretic multiverse | |||
|journal=The Review of Symbolic Logic | |||
|volume=5 |issue=3 |pages=416–449 | |||
|doi=10.1017/S1755020311000359 |s2cid=33807508 |arxiv=1108.4223}} | |||
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| doi = 10.1215/00294527-2835047 | |||
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| journal = Notre Dame Journal of Formal Logic | |||
| title = Is the Dream Solution of the Continuum Hypothesis Attainable? | |||
| volume = 56| arxiv = 1203.4026 | |||
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|doi=10.1090/s0273-0979-03-00981-9 | |||
|arxiv=math/0211398 | |||
|s2cid=1510438 | |||
}} | }} | ||
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|last=Jourdain |first=Philip E.B. | |||
| publisher = Academic Press | |||
|year=1905 | |||
|title=On transfinite cardinal numbers of the exponential form | |||
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|doi=10.1080/14786440509463254 | |||
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|title=Early history of the generalized continuum hypothesis: 1878–1938 | |||
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|doi=10.2307/2695444 | |||
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{{cite journal | |||
|last=Merimovich |first=Carmi | |||
|year=2007 | |||
|title=A power function with a fixed finite gap everywhere | |||
|journal=] | |||
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|doi=10.2178/jsl/1185803615 | |||
|mr=2320282 |arxiv=math/0005179 | |||
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| title = Is the Continuum Hypothesis a definite mathematical problem? | |||
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|title=Zermelo-Fraenkel Set Theory | |||
| title = Axioms of Symmetry: Throwing Darts at the Real Number Line | |||
|location=Columbus, Ohio | |||
| journal = Journal of Symbolic Logic | |||
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| volume = 51 | issue = 1 | year = 1986 | pages = 190–200 | |||
|page=147, exercise 76 | |||
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| first = K. | last = Gödel | |||
| title = The Consistency of the Continuum-Hypothesis | |||
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}} | |||
|title=Believing the axioms, | |||
*{{cite web | |||
|journal=Journal of Symbolic Logic | |||
| first = Peter | last = Koellner | |||
|volume=53 |issue=2 |pages=481–511 | |||
| authorlink = Peter Koellner | |||
|publisher=Association for Symbolic Logic | |||
| url = http://logic.harvard.edu/EFI_CH.pdf | |||
|doi=10.2307/2274520 |jstor=2274520 | |||
| title = The Continuum Hypothesis | |||
}} | |||
| work = Exploring the Frontiers of Independence (Harvard lecture series) | |||
| year = 2011a | |||
==Sources== | |||
}} | |||
* {{PlanetMath attribution|id=1184|title=Generalized continuum hypothesis}} {{Webarchive|url=https://web.archive.org/web/20170208073241/http://planetmath.org/node/31184|date=2017-02-08}} | |||
*{{cite web | |||
| first = Peter | last = Koellner | |||
==Further reading== | |||
| url = http://logic.harvard.edu/EFI_Feferman_comments.pdf | |||
* {{Cite book |last=Cohen |first=Paul Joseph |author-link=Paul Cohen (mathematician) |date=2008 |orig-date=1966 |title=Set theory and the continuum hypothesis |location=Mineola, New York City |publisher=Dover Publications |isbn=978-0-486-46921-8 | |||
| title = Feferman On the Indefiniteness of CH | |||
| year = 2011b <!-- not sure of this, it might be 2012 --> | |||
}} | }} | ||
* {{Cite book |last1=Dales |first1=H.G. |last2=Woodin |first2=W.H. |date= 1987 |title=An Introduction to Independence for Analysts |publisher=Cambridge | |||
*{{Cite book | |||
| last1=Kunen | |||
| first1=Kenneth | |||
| author1-link=Kenneth Kunen | |||
| title=] | |||
| publisher=North-Holland | |||
| location=Amsterdam | |||
| isbn=978-0-444-85401-8 | |||
| year=1980 | |||
| author=Kenneth Kunen. | |||
}} | }} | ||
* {{Cite book |last=Enderton |first=Herbert |date=1977 |title=Elements of Set Theory |publisher=Academic Press | |||
*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 | |||
| first = Penelope | last = Maddy | |||
| journal = Journal of Symbolic Logic | |||
| title = Believing the Axioms, I | |||
| volume = 53 | issue = 2 | month = June | year = 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 | |||
}} | }} | ||
* 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 | |||
* 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}} | |||
| first = Carmi | last = Merimovich | |||
* {{Cite web |author=McGough, Nancy |title=The Continuum Hypothesis |url=http://www.ii.com/math/ch/ }} | |||
| title = A power function with a fixed finite gap everywhere | |||
* {{Cite web |author=Wolchover, Natalie |title=How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer |date=15 July 2021 |url=https://www.quantamagazine.org/how-many-numbers-exist-infinity-proof-moves-math-closer-to-an-answer-20210715/ }} | |||
| journal = Journal of Symbolic Logic | |||
| volume = 72 | issue = 2 | year = 2007 | pages = 361–417 | |||
| doi = 10.2178/jsl/1185803615 | |||
}} | |||
*{{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=]|volume=84|pages=242–258}}. | |||
==External links== | |||
{{PlanetMath attribution|id=1184|title=Generalized continuum hypothesis}} | |||
{{wikiquote-inline}} | |||
*{{MathWorld |title=Continuum Hypothesis |id=ContinuumHypothesis |author=Szudzik, Matthew |author-link=Matthew Szudzik |author2=Weisstein, Eric W. |author2-link=Eric W. Weisstein |name-list-style=amp }} | |||
{{Set theory}} | |||
{{Mathematical logic}} | |||
{{Hilbert's problems}} | {{Hilbert's problems}} | ||
{{Authority control}} | |||
{{DEFAULTSORT:Continuum Hypothesis}} | {{DEFAULTSORT:Continuum Hypothesis}} | ||
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Latest revision as of 23:56, 24 December 2024
Proposition in mathematical logic This article is about the hypothesis in set theory. For the assumption in fluid mechanics, see Continuum assumption. For the album by Epoch of Unlight, see The Continuum Hypothesis (album).
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states:
There is no set whose cardinality is strictly between that of the integers and the real numbers.
Or equivalently:
Any subset of the real numbers is either finite, or countably infinite, or has the cardinality of the real numbers.
In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: , or even shorter with beth numbers: .
The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940.
The name of the hypothesis comes from the term the continuum for the real numbers.
History
Cantor believed the continuum hypothesis to be true and for many years tried in vain to prove it. It became the first on David Hilbert's 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 proved in 1940 that the negation of the continuum hypothesis, i.e., the existence of a set with intermediate cardinality, could not be proved in standard set theory. The second half of the independence of the continuum hypothesis – i.e., unprovability of the nonexistence of an intermediate-sized set – was proved in 1963 by Paul Cohen.
Cardinality of infinite sets
Main article: Cardinal numberTwo 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 integers or rational numbers, the existence of a bijection between two sets becomes more difficult 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 is flawed; it does not take proper account of the fact that all three sets are 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 sets.
Cantor gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers (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 continuum hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. That is, every set, S, of real numbers can either be mapped one-to-one into the integers or the real numbers can be mapped one-to-one into S. As the real numbers are equinumerous with the powerset of the integers, i.e. , the continuum hypothesis can be restated as follows:
Continuum hypothesis — .
Assuming the axiom of choice, there is a unique smallest cardinal number greater than , and the continuum hypothesis is in turn equivalent to the equality .
Independence from ZFC
The independence of the continuum hypothesis (CH) from Zermelo–Fraenkel set theory (ZF) follows from combined work of Kurt Gödel and Paul Cohen.
Gödel showed that CH cannot be disproved from ZF, even if the axiom of choice (AC) is adopted (making ZFC). Gödel's proof shows that CH and AC both hold in the constructible universe L, an inner model of ZF set theory, assuming only the axioms of ZF. The existence of an inner model of ZF in which additional axioms hold shows that the additional axioms are consistent with ZF, provided ZF itself is consistent. The latter condition cannot be proved in ZF itself, due to Gödel's incompleteness theorems, but is widely believed to be true and can be proved in stronger set theories.
Cohen showed that CH cannot be proven from the ZFC axioms, completing the overall independence proof. To prove his result, Cohen developed the method of forcing, which has become a standard tool in set theory. Essentially, this method begins with a model of ZF in which CH holds, and constructs another model which contains more sets than the original, in a way that CH does not hold in the new model. Cohen was awarded the Fields Medal in 1966 for his proof.
The independence proof just described shows that CH is independent of ZFC. Further research has shown that CH is independent of all known large cardinal axioms in the context of ZFC. Moreover, it has been shown that the cardinality of the continuum can be any cardinal consistent with König's theorem. A result of Solovay, proved shortly after Cohen's result on the independence of the continuum hypothesis, shows that in any model of ZFC, if is a cardinal of uncountable cofinality, then there is a forcing extension in which . However, per König's theorem, it is not consistent to assume is or or any cardinal with cofinality .
The continuum hypothesis is closely related to many statements in analysis, point set topology and measure theory. As a result of its independence, many substantial conjectures in those fields have subsequently been shown to be independent as well.
The independence from ZFC means that proving or disproving the CH within ZFC is impossible. However, Gödel and Cohen's negative results are not universally accepted as disposing of all interest in the continuum hypothesis. The continuum hypothesis remains an active topic of research; see Woodin and Peter Koellner for an overview of the current research status.
The continuum hypothesis and the axiom of choice were among the first genuinely mathematical statements shown to be independent of ZF set theory. Although the existence of some statements independent of ZFC had already been known more than two decades prior: for example, assuming good soundness properties and the consistency ZFC, Gödel's incompleteness theorems, which were published in 1931, establish that there is a formal statement (one for each appropriate Gödel numbering scheme) expressing the consistency of ZFC, that is also independent of it. The latter independence result indeed holds for many theories.
Arguments for and against the continuum hypothesis
Gödel believed that CH is false, and that his proof that CH is consistent with ZFC only shows that the Zermelo–Fraenkel axioms do not adequately characterize the universe of sets. Gödel was a Platonist and therefore had no problems with asserting the truth and falsehood of statements independent of their provability. Cohen, though a formalist, also tended towards rejecting CH.
Historically, mathematicians who favored a "rich" and "large" 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.
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.
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 derived by arguing from particular intuitions about probabilities. Freiling believes this axiom is "intuitively clear" but others have disagreed.
A difficult argument against CH developed by W. Hugh Woodin has attracted considerable attention since the year 2000. Foreman does not reject Woodin's argument outright but urges caution. Woodin proposed a new hypothesis that he labeled the "(*)-axiom", or "Star axiom". The Star axiom would imply that is , thus falsifying CH. The Star axiom was bolstered by an independent May 2021 proof showing the Star axiom can be derived from a variation of Martin's maximum. However, Woodin stated in the 2010s that he now instead believes CH to be true, based on his belief in his new "ultimate L" conjecture.
Solomon Feferman argued that CH is not a definite mathematical problem. He proposed 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 suggested that a proposition is mathematically "definite" if the semi-intuitionistic theory can prove . He conjectured that CH is not definite according to this notion, and proposed that CH should, therefore, be considered not to have a truth value. Peter Koellner 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". In a related vein, Saharon Shelah wrote that he does "not agree with the pure Platonic view that the interesting problems in set theory can be decided, that we just have to discover the additional axiom. My mental picture is that we have many possible set theories, all conforming to ZFC".
Generalized continuum hypothesis
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 has the same cardinality as either S or . That is, for any infinite cardinal there is no cardinal such that . GCH is equivalent to:
for every ordinal(occasionally called Cantor's aleph hypothesis).
The beth numbers provide an alternative notation for this condition: for every ordinal . The continuum hypothesis is the special case for the ordinal . GCH was first suggested by Philip Jourdain. For the early history of GCH, see Moore.
Like CH, GCH is also independent of ZFC, but Sierpiński proved that ZF + GCH implies the axiom of choice (AC) (and therefore the negation of the axiom of determinacy, AD), 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 which is smaller than its own Hartogs number—this uses the equality ; for the full proof, see Gillman.
Kurt Gödel showed that GCH is a consequence of ZF + 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 to fail to satisfy . Much later, Foreman and Woodin proved that (assuming the consistency of very large cardinals) it is consistent that holds for every infinite cardinal . Later Woodin extended this by showing the consistency of for every . Carmi Merimovich showed that, for each n ≥ 1, it is consistent with ZFC that for each infinite cardinal κ, 2 is the nth successor of κ (assuming the consistency of some large cardinal axioms). On the other hand, László Patai proved that if γ is an ordinal and for each infinite cardinal κ, 2 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, . If A and B are finite, the stronger inequality 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. GCH implies that for ordinals α and β:
- when α ≤ β+1;
- when β+1 < α and , where cf is the cofinality operation; and
- when β+1 < α and .
The first equality (when α ≤ β+1) follows from:
while:
The third equality (when β+1 < α and ) follows from:
by König's theorem, while:
See also
References
-
Cantor, Georg (1878). "Ein Beitrag zur Mannigfaltigkeitslehre". Journal für die Reine und Angewandte Mathematik. 1878 (84): 242–258. doi:10.1515/crll.1878.84.242 (inactive 1 November 2024).
{{cite journal}}
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This view is often called formalism. Positions more or less like this may be found in Haskell Curry , Abraham Robinson , and Paul Cohen .
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Sources
- This article incorporates material from Generalized continuum hypothesis on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. Archived 2017-02-08 at the Wayback Machine
Further reading
- Cohen, Paul Joseph (2008) . Set theory and the continuum hypothesis. Mineola, New York City: Dover Publications. ISBN 978-0-486-46921-8.
- Dales, H.G.; Woodin, W.H. (1987). An Introduction to Independence for Analysts. Cambridge.
- Enderton, Herbert (1977). Elements of Set Theory. Academic Press.
- 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.
- 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
- McGough, Nancy. "The Continuum Hypothesis".
- Wolchover, Natalie (15 July 2021). "How Many Numbers Exist? Infinity Proof Moves Math Closer to an Answer".
External links
Quotations related to Continuum hypothesis at Wikiquote
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