Cantor showed that the sets of natural numbers and rational numbers could be put into 1-1 correspondence. In some sense, the sets are the same size. (see countable sets)

He also showed that the sets of real numbers and rational numbers can not be put into 1-1 correspondence. In some very real sense there must be more real numbers than rationals. (see uncountable sets)

The Continuum Hypothesis asks if there is an infinity between $\aleph_0,$ the infinity of the Natural Numbers, and $2^{\aleph_0},$ the infinity of the set of all subsets (which is also the infinity of the real numbers).

The answer, surprisingly perhaps, turns out to be "Yes and no."

Paul Cohen showed in 1963 that this question is independent of the usual axioms of set theory (ZFC): in other words, from those axioms it is impossible to prove either that there is, or that there isn't, such a thing as a set bigger than the integers and smaller than the real numbers.

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