# 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:

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

where $$\omega$$ is the first infinite ordinal.