Cardinal

Cardinals are an extension of the natural numbers that describe the size of a set.

There are two ways to define cardinality: cardinals as initial ordinals, or cardinals as equivalence classes under bijectability. The second is more common in settings without the axiom of choice, since not all sets are necessarily well-orderable.^{[Citation needed]}

The aleph numbers are examples of well-ordered cardinals, and exhaust the infinite well-ordered cardinals.Template:Citation nededed

A large cardinal property is often described as a property which states that a cardinal has a certain "largeness" property, such that the existence statement of such a cardinal is unprovable in ZFC. Examples of large cardinal properties include inaccessibility, Mahloness, and indescribability.