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.
