Cardinal number
Cardinal exponentiation

Exponentiation is given by

where X is the set of all functions from Y to X. It is easy to check that the right-hand side depends only on | X | {\displaystyle {|X|}} and | Y | {\displaystyle {|Y|}}.

Exponentiation is non-decreasing in both arguments:

2 is the cardinality of the power set of the set X and Cantor's diagonal argument shows that 2 > | 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). In fact, the class of cardinals is a proper class. (This proof fails in some set theories, notably New Foundations.)

All the remaining propositions in this section assume the axiom of choice:

If 2 ≤ κ and 1 ≤ μ and at least one of them is infinite, then:

Using König's theorem, one can prove κ < κ and κ < cf(2) for any infinite cardinal κ, where cf(κ) is the cofinality of κ.
keywords:forbidden_words, detectable_format:json_format