#bbcode on Rizon

This sequence starts with the natural numbers including zero (finite cardinals), which are followed by the aleph numbers (infinite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs.

Note that 2^|X| is the cardinality of the power set of the set X and Cantor's diagonal argument shows that 2^|X| > |X| for any set X. This proves that no largest cardinal exists (because for any cardinal κ, we can always find a larger cardinal 2[sup]κ[/sup). In fact, the class of cardinals is a proper class. (This proof fails in some set theories, notably New Foundations.)

It is for example presumed impossible for any body to have infinite mass or infinite energy. Concepts of infinite things such as an infinite plane wave exist, but there are no experimental means to generate them.

For example, the empty set is assigned rank 0, while the set {{}} containing only the empty set is assigned rank 1. For each ordinal α, the set Vα is defined to consist of all pure sets with rank less than α. The entire von Neumann universe is denoted V.

The collection of all algebraic objects of a given type will usually be a proper class. Examples include the class of all groups, the class of all vector spaces, and many others. In category theory, a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category.

Additionally, consider for instance the unit circle S, and the action on S by a group G consisting of all rational rotations. Namely, these are rotations by angles which are rational multiples of π. Here G is countable while S is uncountable.

There are several results in category theory which invoke the axiom of choice for their proof. These results might be weaker than, equivalent to, or stronger than the axiom of choice, depending on the strength of the technical foundations.