Continuum hypothesis

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The continuum hypothesis is the hypothesis that there is no set whose cardinality is strictly between that of the integers and that of the real numbers. The real numbers have also been called "the continuum", hence the name.

First conjectured by Cantor, the hypothesis 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 (see set theory). 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.

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 that provably 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 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.

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 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 set theory axioms.


References:

  • The continuum hypothesis
  • 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.
  • H.G.Dales and W.H.Woodin, An Introduction to Independence for Analysts, Cambridge (1987).