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: Main article: History of mathematics The evolution of mathematics can be seen to be an ever increasing series of abstractions. ... Link title As an example, someone who enters a new country and observes only white sheep might form the hypothesis that all sheep in that country are white. ... 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 well-defined collection of 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, aleph usually refers to 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, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one-to-one and onto. ...

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 set is used to describe the size of a set, e. ... 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 If X and Y are sets and every element of X is also an element of Y, then we say or write: X is a subset of (or is included in) Y; X ⊆ Y; Y is a superset of (or includes) X; Y ⊇ X... 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. ...

Investigating the continuum hypothesis

If a set S were found that disproved the continuum hypothesis, it would be impossible to make a one-to-one correspondence between S and the set of integers, because there would always be elements of set S that were "left over". Similarly, it would be impossible to make a one-to-one correspondence between S and the set of real numbers, because there would always be real numbers that were "left over".

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 23 problems in mathematics put forth by German mathematician David Hilbert in the Paris conference of the International Congress of Mathematicians in 1900. ... 1900 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 the Zermelo-Fränkel axiom system and of the axiom of choice. Both of these results assume that the Zermelo-Fränkel axioms themselves do not contain a contradiction, which is widely believed to be true. Kurt Gödel Kurt Gödel [kurt gøːdl], (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 is an axiom of set theory. ... Paul Joseph Cohen (born April 2, 1934) is an American mathematician. ...

As such 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 proved 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 the study of topological spaces. ... In mathematics, a measure is a function that assigns a number, e. ... Mathmatical and Non-Mathamatical Definitions 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. ...

It is interesting to note that Gödel believed strongly that CH is false. To him, his independence of 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. 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. ...

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 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.

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: there are no in-betweens. This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the power set of the integers. GCH is also independent of the Zermelo-Fränkel set theory axioms, and also of the axiom of choice. In mathematics, a set S, the power set of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In mathematics, a set S, the power set of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In mathematics, the axiom of choice is an axiom of set theory. ...

See also

• beth number

In mathematics, the Hebrew letter (aleph) with various subscripts represents various infinite cardinal numbers (see aleph number). ...

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