Continuum hypothesis

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In mathematics, the continuum hypothesis (abbreviated CH) is a hypothesis, advanced by Georg Cantor in 1877, about the possible sizes of infinite sets. It states:

Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's twenty-three 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

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 rationals form a proper superset of the integers, and a proper subset of the reals, so intuitively, there are more rational numbers than integers, and fewer rational numbers than real 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; the second of these is his 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.

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