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In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he showed that the set of integers is strictly smaller than the set of real numbers. The continuum hypothesis states the following: Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...
A hypothesis (foundation from ancient Greek hupothesis where hupo = under and thesis = ) is a proposed explanation for a phenomenon. ...
Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...
In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
Georg Cantor Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 – January 6, 1918) was a mathematician who was born in Russia and lived in Germany for most of his life. ...
In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...
The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
- There is no set whose size is strictly between that of the integers and that of the real numbers.
Or mathematically speaking, noting that the cardinality for the integers  is  ("aleph-null") and the cardinality of the real numbers  is  , the continuum hypothesis says: In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...
In the branch of mathematics known as set theory, the aleph numbers are a series of numbers used to represent the cardinality (or size) of infinite sets. ...
In mathematics, the cardinality of the continuum is the cardinal number of the set of real numbers R (sometimes called the continuum). ...
 This implies:  The real numbers have also been called the continuum, hence the name. There is also a generalization of the continuum hypothesis called the generalized continuum hypothesis saying: In mathematics, the word continuum sometimes denotes the real line. ...
- For all ordinals α:
The size of a set
- Main article: Cardinal number
To state the hypothesis formally, we need a definition: we say that two sets S and T have the same cardinality or cardinal number if there exists a bijection  . Intuitively, this 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}. In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...
In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...
In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...
With infinite sets such as the set of integers or rational numbers, things are more complicated to show. Consider the set of all rational numbers. One might naively suppose that there are more rational numbers than integers, and fewer rational numbers than real numbers, thus disproving the continuum hypothesis. However, it turns out that the rational numbers can be placed in one-to-one correspondence with the integers, and therefore the set of rational numbers is the same size as the set of integers: they are both countable sets. Cantor's diagonal argument shows that the integers and the continuum do not have the same cardinality. The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...
In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
In mathematics the term countable is used to describe the size of a set, i. ...
Cantors diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. ...
The continuum hypothesis states that every subset of the continuum (= the real numbers) which contains the integers either has the same cardinality as the integers or the same cardinality as the continuum. A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
Impossibility of proof and disproof Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. It became the first on David Hilbert's list of important open questions that was presented at the International Mathematical Congress in the year 1900 in Paris. David Hilbert David Hilbert (January 23, 1862–February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ...
Hilberts problems are a list of twenty-three problems in mathematics put forth by German mathematician David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900. ...
1900 (MCM) is a common year starting on Monday. ...
Kurt Gödel showed in 1940 that the continuum hypothesis (CH for short) cannot be disproved from the standard Zermelo-Fraenkel set theory, even if the axiom of choice is adopted. Paul Cohen showed in 1963 that CH cannot be proven from those same axioms either. Hence, CH is independent of ZFC. Both of these results assume that the Zermelo-Fränkel axioms themselves do not contain a contradiction; this assumption is widely believed to be true. Kurt Gödel Kurt Gödel [kurt gøËdl], (April 28, 1906–January 14, 1978) was a logician, mathematician, and philosopher of mathematics. ...
The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...
In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...
Paul Joseph Cohen (born April 2, 1934) is an American mathematician. ...
The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...
It is not surprising that there should be statements which cannot be proven nor disproven within a given axiom system; in fact the content of Gödel's incompleteness theorem is that such statements always exist if the axiom system is strong enough and without contradictions. The independence of CH was still unsettling however, because it was the first concrete example of an important, interesting question of which it could be proven that it could not be decided either way from the universally accepted basic system of axioms on which mathematics is built. In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proven by Kurt Gödel in 1931. ...
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. Analysis is the generic name given to any branch of mathematics which depends upon the concepts of limits and convergence, and studies closely related topics such as continuity, integration, differentiability and transcendental functions. ...
Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...
In mathematics, a measure is a function that assigns a number, e. ...
In mathematics, a conjecture is a mathematical statement which has been proposed as a true statement, but which no one has yet been able to prove or disprove. ...
Arguments pro and con Gödel believed strongly that CH is false. To him, his independence proof only showed that the prevalent set of axioms was defective. Gödel was a platonist and therefore had no problems with asserting truth and falsehood of statements independent of their provability. Cohen, however, was a formalist, but even he tended towards rejecting CH. Platonic idealism is the theory that the substantive reality around us is only a reflection of a higher truth. ...
Historically, mathematicians who favor a "rich" and "large" universe of sets were against CH, while those favoring a "neat" and "controllable" universe favored CH. More recently, some experts (e.g. Matthew Foreman) have pointed out that ontological maximalism can actually be taken as a point in favor of CH, given that between models that have all the same reals, it's the one with more sets of reals that has more chance of satisfying CH. See (Maddy, p. 500). In mathematics, and particularly in applications to set theory and the foundations of mathematics, a universe or universal class (or if a set, universal set) is, roughly speaking, a class that is large enough to contain (in some sense) all of the sets that one may wish to use. ...
Matthew Foreman (March 21, 1957) is a set theorist at University of California, Irvine. ...
Chris Freiling in 1986 presented an argument against CH: he showed that the negation of CH is equivalent to a statement about probabilities which he calls "intuitively true", but others have disagreed. Freilings axiom of symmetry (AX) is a set-theoretic axiom proposed by Chris Freiling. ...
A difficult argument developed by W. Hugh Woodin, against CH, has attracted considerable attention since about the year 2000. See the references in Notices of the AMS. The Foreman reference does not reject Woodin's argument outright but urges caution. W. Hugh Woodin is a set theorist at University of California, Berkeley. ...
The 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 either has the same cardinality as the set S or the same cardinality as the power set of S. That is, for any infinite cardinal λ there is no cardinal κ such that λ < κ < 2λ. An equivalent condition is that  for every ordinal α. Another equivalent condition is that  for every ordinal α. In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...
Infinity is a word carrying a number of different meanings in mathematics, philosophy, theology and everyday life. ...
In linguistics, cardinal numbers is the name given to number words that are used for quantity (one, two, three), as opposed to ordinal numbers, words that are used for order (first, second, third). ...
Commonly, ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...
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 note that ZF + GCH ⊦AC, so that choice and GCH are not independent in ZF; there are no models of ZF in which GCH holds and AC fails. In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ...
References - Cohen, P. J.: Set Theory and the Continuum Hypothesis. New York: W. A. Benjamin, 1966.
- Dales, H. G. and W. H. Woodin: An Introduction to Independence for Analysts. Cambridge (1987).
- Foreman, Matt: Has the Continuum Hypothesis been Settled?
- Freiling, Chris: Axioms of Symmetry: Throwing Darts at the Real Number Line, Journal of Symbolic Logic, Vol. 51, no. 1 (1986), pp. 190-200.
- Gödel, K.: The Consistency of the Continuum-Hypothesis. Princeton, NJ: Princeton University Press, 1940.
- 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.
- Maddy, Penelope: Believing the Axioms, I, Journal of Symbolic Logic, Vol. 53, no. 2 (1988), pp. 481-511.
- McGough, Nancy: The Continuum Hypothesis.
- Woodin, W. Hugh: The Continuum Hypothesis, Part I, Notices of the AMS, Vol. 48, no. 6 (2001), pp. 567-576
- Woodin, W. Hugh: The Continuum Hypothesis, Part II, Notices of the AMS, Vol. 48, no. 7 (2001), pp. 681-690
This article incorporates material from Generalized continuum hypothesis on PlanetMath, which is licensed under the GFDL. In mathematics, the real line is simply the set of real numbers. ...
PlanetMath is a free, collaborative, online mathematics encyclopedia. ...
See also In the branch of mathematics known as set theory, the aleph numbers are a series of numbers used to represent the cardinality (or size) of infinite sets. ...
In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). ...
In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections, injections, and surjections, and another which uses cardinal numbers. ...
External links - The Independence of the Continuum Hypothesis Paul J. Cohen, Proceedings of the National Academy of Sciences of the United States of America, Vol. 50, No. 6. (Dec. 15, 1963), pp. 1143-1148.
- The Independence of the Continuum Hypothesis, II Paul J. Cohen Proceedings of the National Academy of Sciences of the United States of America, Vol. 51, No. 1. (Jan. 15, 1964), pp. 105-110.
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