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Continuum hypothesis

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 (naively: whole numbers) is strictly smaller than the set of real numbers (naively: infinite decimals) 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 <math>|\mathbf{Z}|</math> is <math>\aleph_0</math> and the cardinality for the real numbers <math>|\mathbf{R}|</math> is <math>2^{\aleph_0}</math>, the continuum hypothesis says:

<math>\not\exists A, \aleph_0 < |A| < 2^{\aleph_0}</math>

The real numbers have also been called the continuum, hence the name.

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.

If a set S was 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.

Kurt Gödel showed in 1940 that the continuum hypothesis (CH for short) cannot be disproved from the standard Zermelo-Fraenkel set theory axiom system, 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-Fraenkel axiom system and of the axiom of choice. (Both of these results assume that the Zermelo-Fraenkel axioms themselves don't contain a contradiction, something that's widely believed to be true but impossible to prove.)

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.

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.

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. Nowadays, most researchers in the field are either neutral or reject CH. Generally speaking, mathematicians who favour a "rich" and "large" universe of sets are against CH, while those favoring a "neat" and "controllable" universe favor CH. 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.

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 → 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. 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.

The generalized continuum hypothesis (GCH) states that if a 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 Zermel-Fraenkel set theory axioms and it implies the axiom of choice.

References

  • Nancy McGough.: The Continuum Hypothesis, http://www.ii.com/math/ch/
  • Cohen, P. J.: Set Theory and the Continuum Hypothesis. New York: W. A. Benjamin, 1966.
  • 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.
  • H. G. Dales and W. H. Woodin: An Introduction to Independence for Analysts. Cambridge (1987).
  • Chris Freiling: "Axioms of Symmetry: Throwing Darts at the Real Number Line," Journal of Symbolic Logic, Volume 51 (1986), Issue 1, pp. 190-200.



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