Card

\(\mathrm{Card}\) denotes the class of all cardinals.Stegert, Jan-Carl. "Ordinal proof theory of Kripke-Platek set theory augmented by strong reflection principles" (p.13) It is also commonly denoted as \(\mathrm{CARD}\).Rathjen, Michael. "Ordinal Notations Based on a Weakly Mahlo Cardinal", Archive for Mathematical Logic 29 (1990) 249--263. (p.250)
 denoted it by ת (), the final letter of the The Correspondence between Georg Cantor and Philip Jourdain, I. Grattan-Guinness, Jahresbericht der Deutschen Mathematiker-Vereinigung 73 (1971/72), pp. 111–130, at pp. 116–117.. 
== Properties ==
* The class \(\mathrm{Card}\) is a proper class. If it were a set, it would lead to a paradox of the .
* Assuming the axiom of choice, there is a transfinite sequence of cardinal numbers, starting with the natural numbers, and then followed by the aleph numbers.
** The aleph mapping \(\alpha \rightarrow \aleph_\alpha\) provides a one-to-one correspondance between the ordinals and cardinals. It is the only order-isomorphism between the ordinals and cardinals with respect to membership.
** The aleph numbers are indexed by the ordinal numbers.
* If the axiom of choice does not hold, then there exists infinite cardinals \(\kappa\) that are not aleph numbers.
== Application ==
* Each ordinal is associated with a cardinal. 
** Under the von Neumann definition of ordinals, the cardinality of the ordinal \(\alpha\) is the least ordinal with the same cardinality of \(\alpha\).
** Under this definition, \(\mathrm{Card}\) is a strict subclass of \(\mathrm{On}\).
== See Also ==
== References ==
Category:Set theory
Category:Proper class
Category:Transfinite ordinals