Cardinal
Cardinals (or cardinal numbers) 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. In general, one may informally describe cardinals as numbers, finite or infinite, which are meant to describe how many objects there are in a collection. The cardinality of a set is the (unique) cardinal representing its 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] Here, the cardinality of a set is just the unique equivalence class which the set belongs to. However, in the context of the axiom of 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 \(\Omega\). The cardinality of a set is then defined as the minimal ordinal which it bijects with, although it might require some effort to show that the cardinality of a set is always an initial ordinal.
Using the von Neumann interpretation of ordinals, and the initial ordinal interpretation of cardinals, one gets that any natural number is a cardinal. In particular, under the initial ordinal definition, a cardinal is a natural number if and only if it is finite. Under the equivalence class definition, a cardinal is a natural number if and only if some (furthermore, any) element of it is finite. Again in the initial ordinal definition, every cardinal is an ordinal yet there are many (infinite) ordinals which are not cardinals. However, under the equivalence class definition, no cardinal is an ordinal.
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.
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.