Continuum hypothesis

In mathematics, the continuum hypothesis 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 showed that the set of integers is strictly smaller than the set of real numbers. The continuum hypothesis states the following:

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 $|\mathbb{Z}|$ is $\aleph_0$ ("aleph-null") and the cardinality of the real numbers $|\mathbb{R}|$ is $2^{\aleph_0}$ , the continuum hypothesis says:

\lnot \exists \mathbb{A}: \aleph_0 < |\mathbb{A}| < 2^{\aleph_0}.

This is equivalent to:

2^{\aleph_0} = \aleph_1

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:

For all ordinals

\alpha

2^{\aleph_\alpha} = \aleph_{\alpha+1}

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 $S \leftrightarrow T$ . 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, 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 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.

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.

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-Fraenkel axioms themselves do not contain a contradiction; this assumption is widely believed to be true.

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.

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.

Arguments pro and con

Gödel believed strongly that CH is false. To him, his consistency 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.

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).

Chris Freiling in 1986 presented an argument against CH, called Freiling's axiom of symmetry: he showed that the negation of CH is equivalent to a statement about probabilities which he calls "intuitively true", but others have disagreed.

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.

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 <\kappa <2^{\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 Sierpinski 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).

Implications of GCH for cardinal exponentiation

The Generalized Continuum Hypothesis fixes the value of cardinal exponentiation generally. The value of

\aleph_{\alpha}^{\aleph_{\beta}}

is:

\aleph_{\beta+1}

when α ≤ β+1;

\aleph_{\alpha}

when β+1 < α and the exponent is less than the cofinality of the base; and

\aleph_{\alpha+1}

when β+1 < α and the exponent is greater or equal to the cofinality of the base.

References

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.