section_path: Cardinality
section_name: Cardinality
paragraph_text: Cardinality
  
  The set of all real numbers is uncountable, in the sense that while both the set of all natural numbers {1, 2, 3, 4,...} and the set of all real numbers are infinite sets, there exists no one-to-one function from the real numbers to the natural numbers. The cardinality of the set of all real numbers is denoted by c. {\displaystyle {\mathfrak {c}}.} and called the cardinality of the continuum. It is strictly greater than the cardinality of the set of all natural numbers (denoted ℵ 0 {\displaystyle \aleph _{0}} and called 'aleph-naught'), and equals the cardinality of the power set of the set of the natural numbers.
  
  The statement that there is no subset of the reals with cardinality strictly greater than ℵ 0 {\displaystyle \aleph _{0}} and strictly smaller than c {\displaystyle {\mathfrak {c}}} is known as the continuum hypothesis (CH). It is neither provable nor refutable using the axioms of Zermelo–Fraenkel set theory including the axiom of choice (ZFC)—the standard foundation of modern mathematics. In fact, some models of ZFC satisfy CH, while others violate it.