Editing Cardinal number (section)

=== Cardinal exponentiation ===
Exponentiation is given by
:<math>|X|^{|Y|} = \left|X^Y\right|,</math>
where ''X<sup>Y</sup>'' is the set of all [[function (mathematics)|functions]] from ''Y'' to ''X''.<ref name=":0" /> It is easy to check that the right-hand side depends only on <math>{|X|}</math> and <math>{|Y|}</math>.

:κ<sup>0</sup> = 1  (in particular 0<sup>0</sup> = 1), see [[empty function]].
:If ''μ'' ≥ 1, then 0<sup>''μ''</sup> = 0.
:1<sup>''μ''</sup> = 1.
:''κ''<sup>1</sup> = ''κ''.
:''κ''<sup>''μ'' + ''ν''</sup> = ''κ''<sup>''μ''</sup>·''κ''<sup>''ν''</sup>.
:κ<sup>''μ'' · ''ν''</sup> = (''κ''<sup>''μ''</sup>)<sup>''ν''</sup>.
:(''κ''·''μ'')<sup>''ν''</sup> = ''κ''<sup>''ν''</sup>·''μ''<sup>''ν''</sup>.
Exponentiation is non-decreasing in both arguments:
:(1 ≤ ''ν'' and ''κ'' ≤ ''μ'') → (''ν''<sup>''κ''</sup> ≤ ''ν''<sup>''μ''</sup>) and
:(''κ'' ≤ ''μ'') → (''κ''<sup>''ν''</sup> ≤ ''μ''<sup>''ν''</sup>).

2<sup>|''X''|</sup> is the cardinality of the [[power set]] of the set ''X'' and [[Cantor's diagonal argument]] shows that 2<sup>|''X''|</sup> > |''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 (set theory)|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 ''κ'' and ''μ'' are both finite and greater than 1, and ''ν'' is infinite, then ''κ''<sup>''ν''</sup> = ''μ''<sup>''ν''</sup>.
:If ''κ'' is infinite and ''μ'' is finite and non-zero, then ''κ''<sup>''μ''</sup> = ''κ''.

If 2 ≤ ''κ'' and 1 ≤ ''μ'' and at least one of them is infinite, then:
:Max (''κ'', 2<sup>''μ''</sup>) ≤ ''κ''<sup>''μ''</sup> ≤ Max (2<sup>''κ''</sup>, 2<sup>''μ''</sup>).

Using [[König's theorem (set theory)|König's theorem]], one can prove ''κ'' < ''κ''<sup>cf(''κ'')</sup> and ''κ'' < cf(2<sup>''κ''</sup>) for any infinite cardinal ''κ'', where cf(''κ'') is the [[cofinality]] of ''κ''.

==== Roots ====
Assuming the axiom of choice and, given an infinite cardinal ''κ'' and a finite cardinal ''μ'' greater than 0, the cardinal ''ν'' satisfying <math>\nu^\mu = \kappa</math> will be <math>\kappa</math>.

==== Logarithms ====
Assuming the axiom of choice and, given an infinite cardinal ''κ'' and a finite cardinal ''μ'' greater than 1, there may or may not be a cardinal ''λ'' satisfying <math>\mu^\lambda = \kappa</math>. However, if such a cardinal exists, it is infinite and less than ''κ'', and any finite cardinality ''ν'' greater than 1 will also satisfy <math>\nu^\lambda = \kappa</math>.

The logarithm of an infinite cardinal number ''κ'' is defined as the least cardinal number ''μ'' such that ''κ'' ≤ 2<sup>''μ''</sup>. Logarithms of infinite cardinals are useful in some fields of mathematics, for example in the study of [[cardinal invariant]]s of [[topological space]]s, though they lack some of the properties that logarithms of positive real numbers possess.<ref>Robert A. McCoy and Ibula Ntantu, Topological Properties of Spaces of Continuous Functions, Lecture Notes in Mathematics 1315, [[Springer-Verlag]].</ref><ref>[[Eduard Čech]], Topological Spaces, revised by Zdenek Frolík and Miroslav Katetov, John Wiley & Sons, 1966.</ref><ref>D. A. Vladimirov, Boolean Algebras in Analysis, Mathematics and Its Applications, Kluwer Academic Publishers.</ref>