Continuum hypothesis

In mathematics, the continuum hypothesis is the assumption that there are as many real numbers as there are elements in the smallest set which is larger than the set of natural numbers.

This statement was first made by Georg Cantor when he studied subsets of the real line. Cantor who introduced sets and cardinal numbers, believed this to be true, but tried in vain to prove it.

From then it stayed, for a long time, a prominent open mathematical problem to resolve. In 1900, David Hilbert included the continuum hypothesis as the first problem, therefore also called "first Hilbert problem", in his famous lecture on 23 problems for the twentieth century.

The first step towards a solution was done in 1938 by Kurt Gödel who showed that - in set theory including the axiom of choice - the (generalized) continuum hypothesis cannot be proved to be false (and thus is consistent with it). Only much later, in 1963, J.Paul Cohen showed that it cannot be proved, either. Hence the continuum hypothesis is independent of the usual (ZFC) axioms of set theory. It therefore constitutes an important, not artificially constructed, example for Gödel's Second Incompleteness Theorem.

Consequently, either the continuum hypothesis or, alternatively, some contradicting assumption could be added to axioms of set theory. But since - in contrast to the situation with the axiom of choice - there is no heuristically convincing reason to choose one of these possibilities, the "working" mathematician usually makes no use of the continuum hypotheses, and if a result depends on it, then it is explicitly mentioned.

Of course, in axiomatic set theory, and especially in the theory of cardinal and ordinal numbers, the situation is different and the consequences of the various choices concerning the continuum hypothesis are extensively studied.

The generalized continuum hypothesis is a much stronger statement involving the initial sequence of transfinite cardinal numbers, and is also independent of ZFC.
In terms of the arithmetic of cardinal numbers (as introduced by Cantor) the continuum hypothesis reads

$\aleph _{0}<2^{\aleph _{0}}$

while the generalized continuum hypothesis is

$\aleph _{n}<2^{\aleph _{n}}(\forall n\in \mathbb {N} )$

David Hilbert 1900

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