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The reals are uncountable; that is: while both the set of all natural numbers and the set of all real numbers are infinite sets, there can be no one-to-one function from the real numbers to the natural numbers: the cardinality of the set of all real numbers (denoted ${\displaystyle {\mathfrak {c}}}$ and called cardinality of the continuum) is strictly greater than the cardinality of the set of all natural numbers (denoted ${\displaystyle \aleph _{0}}$ 'aleph-naught'). The statement that there is no subset of the reals with cardinality strictly greater than ${\displaystyle \aleph _{0}}$ and strictly smaller than ${\displaystyle {\mathfrak {c}}}$ is known as the continuum hypothesis (CH). It is known to be neither provable nor refutable using the axioms of Zermelo–Fraenkel set theory including the