Cardinal
Cardinals are an extension of the natural numbers that describe the size of a set. Unlike ordinals, which describe "length" and order, cardinals are only intended to describe size.
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.[1] However, in the context of axiom choice, the former is more common because the objects we work with are sets rather than proper classes. In particular, a cardinal is just defined as an ordinal which does not biject with any smaller ordinal. All finite ordinals are cardinals, as well as \(\omega\). The next cardinal after \(\omega\) is \(\omega_1\), aka \(\Omega\).
Typically, the \(\alpha\)th well-ordered cardinal is denoted by \(\aleph_\alpha\). For example, in the context of choice, the least infinite cardinal is \(\aleph_0\) - in the context of initial ordinals, this is used interchangeably to mean \(\omega\). The next cardinal is \(\aleph_1\), which is used interchangeably with \(\omega_1\).
If choice doesn't outright fail, one may talk about well-ordered and non-well ordered cardinals. Aleph numbers are examples of well-ordered cardinals, and exhaust the infinite well-ordered cardinals. The existence of cardinals such as \(\aleph_\omega\) is guaranteed by the axiom of replacement, and these can not exist without the axiom of replacement, since one isn't in general able to take the suprema of arbitrary sets.
A large cardinal axiom 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.[2] Examples of large cardinal properties include inaccessibility, Mahloness, and indescribability.