 Cardinal number

A cardinal number records the size of a set up to bijection. Two sets have the same cardinality exactly when such a bijection exists. Finite cardinals are $0,1,2,\dots$; the least infinite one is ${\aleph }_0$, the size of the natural numbers.

With the axiom of choice, every set can be well-ordered, and its cardinal is represented by the least ordinal in its bijection class, an initial ordinal. Infinite size does not behave like finite counting: the even natural numbers and all natural numbers have equal cardinality via $n\mapsto 2n$. This notion is distinct from the order type recorded by an ordinal.

A cardinal number is a number that expresses quantity. It tells us "how many" of something there are. For example, the numbers 1, 2, 3, and so on are cardinal numbers because they indicate specific counts of objects. Cardinal numbers can be finite (like 0, 1, 2, 3) or infinite (like the concept of infinity).