Continuum hypothesis

This article is about a hypothesis in set theory. For the assumption in fluid mechanics, see fluid mechanics. Fluid mechanics is the study of how Fluids move and the Forces on them

In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis, advanced by Georg Cantor, about the possible sizes of infinite sets. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A hypothesis (from Greek) consists either of a suggested explanation for a phenomenon (an event that is observable or of a reasoned proposal suggesting a possible Georg Ferdinand Ludwig Philipp Cantor ( – January 6 1918) was a German Mathematician, born in Russia. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness Cantor introduced the concept of cardinality to compare the sizes of infinite sets, and he gave two proofs that the cardinality of the set of integers is strictly smaller than that of the set of real numbers. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, the real numbers may be described informally in several different ways His proofs, however, give no indication of the extent to which the cardinality of the natural numbers is less than that of the real numbers. Cantor proposed the continuum hypothesis as a possible solution to this question. It states:

There is no set whose size is strictly between that of the integers and that of the real numbers.

In light of Cantor's theorem that the sizes of these sets cannot be equal, this hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. The name of the hypothesis comes from the term the continuum for the real numbers. In Mathematics, the word continuum has at least two distinct meanings outlined in the sections below

Equivalently, as the cardinality of the integers is $\aleph_0$ ("aleph-null") and the cardinality of the real numbers is $2^{\aleph_0}$ , the continuum hypothesis says that there is no set $S$ for which $\aleph_0 < |S| < 2^{\aleph_0}.$ Assuming the axiom of choice, there is a smallest cardinal number $\aleph_1$ greater than $\aleph_0$ , and the continuum hypothesis is in turn equivalent to the equality

$2^{\aleph_0} = \aleph_1.$

There is also a generalization of the continuum hypothesis called the generalized continuum hypothesis (GCH) saying:

For all ordinals

α

, $2^{\aleph_\alpha} = \aleph_{\alpha+1}.$

1 As the first Hilbert problem
2 The size of a set
3 Impossibility of proof and disproof (in ZFC)
4 Arguments for and against CH
5 The generalized continuum hypothesis
- 5.1 Implications of GCH for cardinal exponentiation
6 See also
7 References

As the first Hilbert problem

In 1900, David Hilbert posed the question of whether the continuum hypothesis holds; it was the first of the celebrated Hilbert problems. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. In Mathematics, the cardinality of the continuum, sometimes also called the power of the continuum, is the size ( Cardinality) of the set of In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Hilbert's problems are a list of twenty-three problems in Mathematics put forth by German Mathematician David Hilbert at the Paris Later work by Kurt Gödel in 1939 showed that the continuum hypothesis could not be disproved based on the current axioms of set theory (ZFC). Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher In 1963 Paul Cohen established that the continuum hypothesis is not provable under the Zermelo-Fraenkel set theory axioms with choice (Enderton 1977). Paul Joseph Cohen ( April 2, 1934 &ndash March 23, 2007) was an American Mathematician best known for his proof of Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common

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).

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 between S and T. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property 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}.

With infinite sets such as the set of integers or rational numbers, this becomes more complicated to demonstrate. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions 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. Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891

The continuum hypothesis states that every infinite subset of the continuum (the real numbers) either has the same cardinality as the integers or the same cardinality as the continuum. In Mathematics, the real numbers may be described informally in several different ways

Impossibility of proof and disproof (in ZFC)

Cantor believed the continuum hypothesis to be true and tried for many years to prove it, in vain. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true 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. The International Congress of Mathematicians (ICM is the largest congress in the Mathematics community Year 1900 ( MCM) was an exceptional Common year starting on Monday (link will display the full calendar of the Gregorian calendar Axiomatic set theory was at that point not yet formulated.

Kurt Gödel showed in 1940 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). Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. Paul Cohen showed in 1963 that CH cannot be proven from those same axioms either. Paul Joseph Cohen ( April 2, 1934 &ndash March 23, 2007) was an American Mathematician best known for his proof of Hence, CH is independent of ZFC. In Mathematical logic, a sentence &sigma is called independent of a given first-order theory T if T neither proves nor Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common 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 expressing the consistency of ZFC that is independent of ZFC. In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most This consistency statement is of a metamathematical, rather than purely mathematical, character. The continuum hypothesis and the axiom of choice were among the first mathematical statements shown to be independent of ZF set theory. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. These independence proofs were not completed until Paul Cohen developed forcing in the 1960s. In the mathematical discipline of Set theory, forcing is a technique invented by Paul Cohen, for proving Consistency and independence results

The continuum hypothesis is closely related to many statements in analysis, point set topology and measure theory. Analysis has its beginnings in the rigorous formulation of Calculus. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with As a result of its independence, many substantial conjectures in those fields have subsequently been shown to be independent as well. In Mathematics, a conjecture is a Mathematical statement which appears resourceful but has not been formally proven to be true under the rules of

So far, CH appears to be independent of all known large cardinal axioms in the context of ZFC. In the mathematical field of Set theory, a large cardinal property is a certain kind of property of Transfinite Cardinal numbers Cardinals with such properties

Arguments for and against CH

Gödel believed that CH is false and that his proof that CH is consistent only shows that the Zermelo-Frankel axioms are defective. Gödel was a platonist and therefore had no problems with asserting the truth and falsehood of statements independent of their provability. The philosophy of mathematics is the branch of Philosophy that studies the philosophical assumptions foundations and implications of Mathematics. Cohen, though a formalist, also tended towards rejecting CH. The philosophy of mathematics is the branch of Philosophy that studies the philosophical assumptions foundations and implications of Mathematics.

Historically, mathematicians who favored a "rich" and "large" universe of sets were against CH, while those favoring a "neat" and "controllable" universe favored CH. In Mathematical logic, the universe of a structure (or model) is its domain. Parallel arguments were made for and against the axiom of constructibility, which implies CH. The axiom of constructibility is a possible Axiom for Set theory in mathematics that asserts that every set is constructible. 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. Matthew Foreman (born March 21, 1957) is a set theorist at University of California Irvine. In Philosophy, ontology (from the Greek, genitive: of being (part For the Marxist concept see Maximum programme. For the theological concept see Biblical maximalism. 500).

Another viewpoint is that the naive conception of set is not specific enough to determine whether CH is true or false. This viewpoint is 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 about probabilities. Freiling's axiom of symmetry ( AX) is a set-theoretic axiom proposed by Chris Freiling. Probability is the likelihood or chance that something is the case or will happen 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). William Hugh Woodin (b April 23, 1955, Tucson, Arizona) is a set theorist at University of California Berkeley. Foreman (2003) 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. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) That is, for any infinite cardinal $λ$ there is no cardinal $κ$ such that $λ < κ < 2 λ . Infinity (symbolically represented with \infty) comes from the Latin infinitas or "unboundedness This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English.$ An equivalent condition is that $\aleph_{\alpha+1}=2^{\aleph_\alpha}$ for every ordinal $α. In Set theory, an ordinal number, or just ordinal, is the Order type of a Well-ordered set.$ The beth numbers provide an alternate notation for this condition: $\aleph_\alpha=\beth_\alpha$ for every ordinal $α. In Mathematics, the Infinite Cardinal numbers are represented by the Hebrew letter \aleph ( aleph) indexed with a subscript that runs$

This is a generalization of the continuum hypothesis since the continuum has the same cardinality as the power set of the integers. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) Like CH, GCH is also independent of ZFC, but 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. Wacław Franciszek Sierpiński ( March 14 1882 — October 21 1969) (ˈvaʦwaf fraɲˈʨiʂɛk ɕɛrˈpʲiɲskʲi a Polish Mathematician In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.

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 consistent with ZFC. Kurt Gödel (kʊɐ̯t ˈgøːdl̩ (April 28 1906 – January 14 1978 was an Austrian American Logician, Mathematician and Philosopher The axiom of constructibility is a possible Axiom for Set theory in mathematics that asserts that every set is constructible. 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 $\aleph_\alpha$ to fail to satisfy $2^{\aleph_\alpha} = \aleph_{\alpha + 1}$ . In Set theory, Easton's theorem is a result on the possible Cardinal numbers of Powersets W

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,

$A < B \to 2^A \le 2^B$ .

If A and B are finite, the stronger inequality

$A < B \to 2^A < 2^B \!$

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 $\aleph_{\alpha}^{\aleph_{\beta}}$ is:

$\aleph_{\beta+1}$ when α ≤ β+1;

$\aleph_{\alpha}$ when β+1 < α and $\aleph_{\beta} < \operatorname{cf} (\aleph_{\alpha})$ where cf is the cofinality operation; and

$\aleph_{\alpha+1}$ when β+1 < α and $\aleph_{\beta} \ge \operatorname{cf} (\aleph_{\alpha})$ . In Mathematics, especially in Order theory, the cofinality cf( A) of a Partially ordered set A is the least of the cardinalities

References

Cohen, P. In Mathematics, the Infinite Cardinal numbers are represented by the Hebrew letter \aleph ( aleph) indexed with a subscript that runs In Mathematics, the cardinality of a set is a measure of the "number of elements of the set" J. (1966). Set Theory and the Continuum Hypothesis. W. A. Benjamin.
Cohen, Paul J. (December 15, 1963). "The Independence of the Continuum Hypothesis". Proceedings of the National Academy of Sciences of the United States of America 50 (6): 1143–1148.
Cohen, Paul J. (January 15, 1964). "The Independence of the Continuum Hypothesis, II". Proceedings of the National Academy of Sciences of the United States of America 51 (1): 105–110.
Dales, H. G. ; W. H. Woodin (1987). An Introduction to Independence for Analysts. Cambridge.
Enderton, Herbert (1977). Elements of Set Theory. Academic Press.
Foreman, Matt (2003). Has the Continuum Hypothesis been Settled? (PDF). Retrieved on February 25, 2006.
Freiling, Chris (1986). "Axioms of Symmetry: Throwing Darts at the Real Number Line". Journal of Symbolic Logic 51 (1): 190–200.
Gödel, K. (1940). The Consistency of the Continuum-Hypothesis. Princeton University 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.
Maddy, Penelope (June 1988). "Believing the Axioms, I". Journal of Symbolic Logic 53 (2): 481–511.
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.
Woodin, W. Hugh (2001a). "The Continuum Hypothesis, Part I". Notices of the AMS 48 (6): 567–576.
Woodin, W. Hugh (2001b). "The Continuum Hypothesis, Part II". Notices of the AMS 48 (7): 681–690.

This article incorporates material from Generalized continuum hypothesis on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative online Mathematics Encyclopedia.

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