Definition:Cardinal
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Definition
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:
- $\operatorname{Card} \left({S}\right) \sim S$, i.e. $\operatorname{Card} \left({S}\right)$ is (set) equivalent to $S$
- $S \sim T \iff \operatorname{Card} \left({S}\right) = \operatorname{Card} \left({T}\right)$.
Notes
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.
