Editing Cardinal number (section)

== Cardinal arithmetic ==
We can define [[arithmetic]] operations on cardinal numbers that generalize the ordinary operations for natural numbers.  It can be shown that for finite cardinals, these operations coincide with the usual operations for natural numbers. Furthermore, these operations share many properties with ordinary arithmetic.

=== Successor cardinal ===
{{Further|Successor cardinal}}

If the axiom of choice holds, then every cardinal κ has a successor, denoted κ<sup>+</sup>, where κ<sup>+</sup> > κ and there are no cardinals between κ and its successor.  (Without the axiom of choice, using [[Hartogs number|Hartogs' theorem]], it can be shown that for any cardinal number κ, there is a minimal cardinal κ<sup>+</sup> such that <!-- κ<sup>+</sup> ࣞ κ.<ref group=notes>The symbol is the [[Unicode]] symbol for not less than or equal to.</ref>--><math>\kappa^+\nleq\kappa. </math>)  For finite cardinals, the successor is simply κ + 1.  For infinite cardinals, the successor cardinal differs from the [[successor ordinal]].

=== Cardinal addition ===
If ''X'' and ''Y'' are [[Disjoint sets|disjoint]], addition is given by the [[union (set theory)|union]] of ''X'' and ''Y''.  If the two sets are not already disjoint, then they can be replaced by disjoint sets of the same cardinality (e.g., replace ''X'' by ''X''×{0} and ''Y'' by ''Y''×{1}).
:<math>|X| + |Y| = | X \cup Y|.</math><ref name=":0">{{harvnb|Schindler|2014|loc=pg. 34}}</ref>

Zero is an additive identity ''κ'' + 0 = 0 + ''κ'' = ''κ''.

Addition is [[associative]] (''κ'' + ''μ'') + ''ν'' = ''κ'' + (''μ'' + ''ν'').

Addition is [[commutative]] ''κ'' + ''μ'' = ''μ'' + ''κ''.

Addition is non-decreasing in both arguments:
:<math>(\kappa \le \mu) \rightarrow ((\kappa + \nu \le \mu + \nu) \mbox{ and } (\nu + \kappa \le \nu + \mu)).</math>

Assuming the axiom of choice, addition of infinite cardinal numbers is easy.  If either ''κ'' or ''μ'' is infinite, then
:<math>\kappa + \mu = \max\{\kappa, \mu\}\,.</math>

==== Subtraction ====
Assuming the axiom of choice and, given an infinite cardinal ''σ'' and a cardinal ''μ'', there exists a cardinal ''κ'' such that ''μ'' + ''κ'' = ''σ'' if and only if ''μ'' ≤ ''σ''. It will be unique (and equal to ''σ'') if and only if ''μ'' < ''σ''.

=== Cardinal multiplication ===
The product of cardinals comes from the [[Cartesian product]].
:<math>|X|\cdot|Y| = |X \times Y|</math><ref name=":0" />

Zero is a multiplicative [[absorbing element]]: ''κ''·0 = 0·''κ'' = 0.

There are no nontrivial [[zero divisor]]s: ''κ''·''μ'' = 0 → (''κ'' = 0 or ''μ'' = 0).

One is a multiplicative identity: ''κ''·1 = 1·''κ'' = ''κ''.

Multiplication is associative: (''κ''·''μ'')·''ν'' = ''κ''·(''μ''·''ν'').

Multiplication is [[commutative]]: ''κ''·''μ'' = ''μ''·''κ''.

Multiplication is non-decreasing in both arguments:
''κ'' ≤ ''μ'' → (''κ''·''ν'' ≤ ''μ''·''ν'' and ''ν''·''κ'' ≤ ''ν''·''μ'').

Multiplication [[distributivity|distributes]] over addition:
''κ''·(''μ'' + ''ν'') = ''κ''·''μ'' + ''κ''·''ν'' and
(''μ'' + ''ν'')·''κ'' = ''μ''·''κ'' + ''ν''·''κ''.

Assuming the axiom of choice, multiplication of infinite cardinal numbers is also easy. If either ''κ''  or ''μ'' is infinite and both are non-zero, then
:<math>\kappa\cdot\mu = \max\{\kappa, \mu\}.</math>
Thus the product of two infinite cardinal numbers is equal to their sum.

==== Division ====
Assuming the axiom of choice and given an infinite cardinal ''π'' and a non-zero cardinal ''μ'', there exists a cardinal ''κ'' such that ''μ'' · ''κ'' = ''π'' if and only if ''μ'' ≤ ''π''. It will be unique (and equal to ''π'') if and only if ''μ'' < ''π''.

=== 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>