# cardinal

A *cardinal* number represents the size of a set irrespective of the order or structure of its elements. If \(X\) is a set, its *cardinality* is generally indicated by \(\left| X\right|\). Two sets \(A\) and \(B\) are said to have the same cardinality if there is a function from \(A\) into \(B\) that is bijective (i.e., one-to-one and onto).

A cardinal is an initial ordinal, i.e., an ordinal for which there does not exist a bijection onto any lesser ordinal. Thus, all finite ordinals (i.e., the natural numbers) are cardinals, but most transfinite ordinals are not cardinals.

If \(a\) is a cardinal, then we denote by \(a^+\) the least cardinal greater than \(a\). A cardinal \(\kappa\) is called a *successor cardinal* if \(\kappa = a^+\) for some cardinal \(a\). Otherwise \(\kappa\) is called a *limit cardinal*. The heirarchy of transfinite cardinals is defined by recursion, and denoted using the Hebrew letter aleph:

- \(\aleph_0 = \left| \omega\right| \)
- \(\aleph_{\alpha + 1} = \left( \aleph_{\alpha} \right)^+\)
- if \(\gamma\) is a limit cardinal, then \(\aleph_{\gamma}=\sup\left\{ \aleph_{\alpha}:\alpha < \gamma \right\} \)

where \(\omega\) is the first infinite ordinal.