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Let $S$ be a set.

Associated with $S$ there exists a set $\operatorname{Card} \left({S}\right)$ called the cardinal of $S$.

It has the properties:

$(1): \quad \map \Card S \sim S$

that is, $\map \Card S$ is (set) equivalent to $S$

$(2): \quad S \sim T \iff \map \Card S = \map \Card T$


A cardinal is an equivalence class of sets of the same cardinality, so each cardinal has no specific nature as a set.

The cardinals do not form a set, since this would be $\mathcal S / \sim$, where $\mathcal S$ is the set of all sets, which by Set of all Sets leads to a contradiction.

See Cardinality for further discussion on the subject.

Also see

  • Results about cardinals can be found here.