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Continuum hypothesis

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In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis, advanced by Georg Cantor in 1878, about the possible sizes of infinite sets. It states:

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 Hilbert's 23 problems presented in the year 1900. The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo-Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent.

The name of the hypothesis comes from the term the continuum for the real numbers.

Cardinality of infinite sets

Main article: Cardinal number

Two 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, this becomes more complicated 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 does not take 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 hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. Equivalently, as the cardinality of the integers is ℵ 0 {\displaystyle \aleph _{0}} ("aleph-naught") and the cardinality of the real numbers is 2 ℵ 0 {\displaystyle 2^{\aleph _{0}}} , the continuum hypothesis says that there is no set S {\displaystyle S} for which

ℵ 0 < | S | < 2 ℵ 0 . {\displaystyle \aleph _{0}<|S|<2^{\aleph _{0}}.\,}

Assuming the axiom of choice, there is a smallest cardinal number ℵ 1 {\displaystyle \aleph _{1}} greater than ℵ 0 {\displaystyle \aleph _{0}} , and the continuum hypothesis is in turn equivalent to the equality

2 ℵ 0 = ℵ 1 . {\displaystyle 2^{\aleph _{0}}=\aleph _{1}.\,}

There is also a generalization of the continuum hypothesis called the generalized continuum hypothesis (GCH) which says that for all ordinals α {\displaystyle \alpha \,}

2 ℵ α = ℵ α + 1 . {\displaystyle 2^{\aleph _{\alpha }}=\aleph _{\alpha +1}.}

A consequence of the hypothesis is that every infinite subset of the real numbers either has the same cardinality as the integers or the same cardinality as the entire set of the reals.

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 (one for each appropriate Gödel numbering scheme) expressing the consistency of ZFC that is independent of ZFC. 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.

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.

Arguments for and against CH

Gödel believed that CH is false and that his proof that CH is consistent only shows that the Zermelo-Fraenkel axioms do not adequately describe 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 (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 (Kunen 1980, p. 171).

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.

Solomon Feferman (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 classical logic for bounded quantifiers but uses intuitionistic logic for unbounded ones, and suggests that a proposition ϕ {\displaystyle \phi } is mathematically "definite" if the semi-intuitionistic theory can prove ( ϕ ∨ ¬ ϕ ) {\displaystyle (\phi \lor \neg \phi )} . 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

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 {\displaystyle \lambda \,} there is no cardinal κ {\displaystyle \kappa \,} such that λ < κ < 2 λ . {\displaystyle \lambda <\kappa <2^{\lambda }.\,} An equivalent condition is that ℵ α + 1 = 2 ℵ α {\displaystyle \aleph _{\alpha +1}=2^{\aleph _{\alpha }}} for every ordinal α . {\displaystyle \alpha .\,} The beth numbers provide an alternate notation for this condition: ℵ α = ℶ α {\displaystyle \aleph _{\alpha }=\beth _{\alpha }} for every ordinal α . {\displaystyle \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 ℵ α {\displaystyle \aleph _{\alpha }} to fail to satisfy 2 ℵ α = ℵ α + 1 . {\displaystyle 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 κ > κ + {\displaystyle 2^{\kappa }>\kappa ^{+}\,} holds for every infinite cardinal κ . {\displaystyle \kappa .\,} Later Woodin extended this by showing the consistency of 2 κ = κ + + {\displaystyle 2^{\kappa }=\kappa ^{++}\,} for every κ {\displaystyle \kappa \,} . A recent result of Carmi Merimovich shows that, for each n≥1, it is consistent with ZFC that for each κ, 2 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, 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,

A < B → 2 A ≤ 2 B {\displaystyle A<B\to 2^{A}\leq 2^{B}} .

If A and B are finite, the stronger inequality

A < B → 2 A < 2 B {\displaystyle 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 ℵ α ℵ β {\displaystyle \aleph _{\alpha }^{\aleph _{\beta }}} is:

ℵ β + 1 {\displaystyle \aleph _{\beta +1}} when α ≤ β+1;
ℵ α {\displaystyle \aleph _{\alpha }} when β+1 < α and ℵ β < cf ⁡ ( ℵ α ) {\displaystyle \aleph _{\beta }<\operatorname {cf} (\aleph _{\alpha })} where cf is the cofinality operation; and
ℵ α + 1 {\displaystyle \aleph _{\alpha +1}} when β+1 < α and ℵ β ≥ cf ⁡ ( ℵ α ) {\displaystyle \aleph _{\beta }\geq \operatorname {cf} (\aleph _{\alpha })} .

See also

References

Primary literature in German

This article incorporates material from Generalized continuum hypothesis on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Hilbert's problems
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