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paired off

Continuum hypothesis

In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis, advanced by Georg Cantor, about the possible sizes of infinite sets. 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. 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.

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 alpha, 2^{aleph_alpha} = aleph_{alpha+1}.

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. Later work by Kurt Gödel in 1939 showed that the continuum hypothesis could not be disproved based on the current axioms of set theory (ZF). In 1963 Paul Cohen established that the continuum hypothesis is not provable under the Zermelo-Fraenkel set theory axioms with choice (ZFC) (Enderton 1977).

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

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. 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. 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 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.

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. 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 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). 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-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. 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. These independence proofs were not completed until Paul Cohen developed forcing in the 1960s.

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.

So far, CH appears to be independent of all known large cardinal axioms in the context of ZFC.

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. 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 (Maddy 1988, p. 500).

Another viewpoint is that the conception of set is not specific enough to determine whether CH is true or false. This viewpoint was advanced as early as 1923 by Skolem, even before Gödel's first incompleteness theorem. Skolem argued on the basis of what is now known as Skolem's paradox, and it was later supported by the independence of CH from the axioms of ZFC, since these axioms are enough to establish the elementary properties of sets and cardinalities. In order to argue against this viewpoint, it would be sufficient to demonstrate new axioms that are supported by intuition and resolve CH in one direction or another. Although the axiom of constructibility does resolve CH, it is not generally considered to be intuitively true any more than CH is generally considered to be false.

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 believes this axiom is "intuitively true" but others have disagreed. A difficult argument against CH developed by W. Hugh Woodin has attracted considerable attention since the year 2000 (Woodin 2001a, 2001b). Foreman (2003) does not reject Woodin's argument outright but urges caution.

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 lambda there is no cardinal kappa such that lambda An equivalent condition is that aleph_{alpha+1}=2^{aleph_alpha} for every ordinal alpha. The beth numbers provide an alternate notation for this condition: aleph_alpha=beth_alpha for every ordinal alpha.

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 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.

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. 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}. Much later, Foreman and Woodin proved that (assuming the consistency of very large cardinals) it is consistent that 2^kappa>kappa^+ holds for every infinite cardinal kappa. Later Woodin extended this by showing the consistency of 2^kappa=kappa^{++} for every kappa. A recent result of Carmi Merimovich shows that, for each n≥1, it is consistent with ZFC that for each κ, 2κ is the nth successor of κ. On the other hand, Laszlo Patai proved, that if γ is an ordinal and for each infinite cardinal κ, 2κ 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,

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}).

See also

References

  • Cohen, P. J. (1966). Set Theory and the Continuum Hypothesis. W. A. Benjamin.
  • Cohen, Paul J. "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. "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 Has the Continuum Hypothesis been Settled?. (2003). .
  • 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 (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. .
  • Merimovich, Carmi (2007). "A power function with a fixed finite gap everywhere". Journal of Symbolic Logic 72 (2): 361–417.
  • 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.

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