cardinal arithmetic

Definitions

Let $\kappa$ and $\lambda$ be cardinal numbers, and let $A$ and $B$ be disjoint sets such that $|A|=\kappa$ and $|B|=\lambda$ . (Here $|X|$ denotes the cardinality of a set $X$ , that is, the unique cardinal number equinumerous with $X$ .) Then we define cardinal addition, cardinal multiplication and cardinal exponentiation as follows.

	$\displaystyle\kappa+\lambda$	$\displaystyle=\|A\cup B\|.$
	$\displaystyle\kappa\lambda$	$\displaystyle=\|A\times B\|.$
	$\displaystyle\kappa^{\lambda}$	$\displaystyle=\|A^{B}\|.$

(Here $A^{B}$ denotes the set of all functions from $B$ to $A$ .) These three operations are well-defined, that is, they do not depend on the choice of $A$ and $B$ . Also note that for multiplication and exponentiation $A$ and $B$ do not actually need to be disjoint.

We also define addition and multiplication for arbitrary numbers of cardinals. Suppose $I$ is an index set and $\kappa_{i}$ is a cardinal for every $i\in I$ . Then $\sum_{i\in I}\kappa_{i}$ is defined to be the cardinality of the union $\bigcup_{i\in I}A_{i}$ , where the $A_{i}$ are pairwise disjoint and $|A_{i}|=\kappa_{i}$ for each $i\in I$ . Similarly, $\prod_{i\in I}\kappa_{i}$ is defined to be the cardinality of the Cartesian product (http://planetmath.org/GeneralizedCartesianProduct) $\prod_{i\in I}B_{i}$ , where $|B_{i}|=\kappa_{i}$ for each $i\in I$ .

Properties

In the following, $\kappa$ , $\lambda$ , $\mu$ and $\nu$ are arbitrary cardinals, unless otherwise specified.

Cardinal arithmetic obeys many of the same algebraic laws as real arithmetic. In particular, the following properties hold.

	$\displaystyle\kappa+\lambda$	$\displaystyle=\lambda+\kappa.$
	$\displaystyle(\kappa+\lambda)+\mu$	$\displaystyle=\kappa+(\lambda+\mu).$
	$\displaystyle\kappa\lambda$	$\displaystyle=\lambda\kappa.$
	$\displaystyle(\kappa\lambda)\mu$	$\displaystyle=\kappa(\lambda\mu).$
	$\displaystyle\kappa(\lambda+\mu)$	$\displaystyle=\kappa\lambda+\kappa\mu.$
	$\displaystyle\kappa^{\lambda}\kappa^{\mu}$	$\displaystyle=\kappa^{\lambda+\mu}.$
	$\displaystyle(\kappa^{\lambda})^{\mu}$	$\displaystyle=\kappa^{\lambda\mu}.$
	$\displaystyle\kappa^{\mu}\lambda^{\mu}$	$\displaystyle=(\kappa\lambda)^{\mu}.$

Some special cases involving $0$ and $1$ are as follows:

	$\displaystyle\kappa+0$	$\displaystyle=\kappa.$
	$\displaystyle 0\kappa$	$\displaystyle=0.$
	$\displaystyle\kappa^{0}$	$\displaystyle=1.$
	$\displaystyle 0^{\kappa}$	$\displaystyle=0,\text{ for }\kappa>0.$
	$\displaystyle 1\kappa$	$\displaystyle=\kappa.$
	$\displaystyle\kappa^{1}$	$\displaystyle=\kappa.$
	$\displaystyle 1^{\kappa}$	$\displaystyle=1.$

If at least one of $\kappa$ and $\lambda$ is infinite, then the following hold.

	$\displaystyle\kappa+\lambda$	$\displaystyle=\max(\kappa,\lambda).$
	$\displaystyle\kappa\lambda$	$\displaystyle=\max(\kappa,\lambda),\text{ provided }\kappa\neq 0\neq\lambda.$

Also notable is that if $\kappa$ and $\lambda$ are cardinals with $\lambda$ infinite and $2\leq\kappa\leq 2^{\lambda}$ , then

\displaystyle\kappa^{\lambda}

\displaystyle=2^{\lambda}.

Inequalities are also important in cardinal arithmetic. The most famous is Cantor’s theorem

\displaystyle\kappa

\displaystyle<2^{\kappa}.

If $\mu\leq\kappa$ and $\nu\leq\lambda$ , then

	$\displaystyle\mu+\nu$	$\displaystyle\leq\kappa+\lambda.$
	$\displaystyle\mu\nu$	$\displaystyle\leq\kappa\lambda.$
	$\displaystyle\mu^{\nu}$	$\displaystyle\leq\kappa^{\lambda},\text{ unless }\mu=\nu=\kappa=0<\lambda.$

Similar inequalities hold for infinite sums and products. Let $I$ be an index set, and suppose that $\kappa_{i}$ and $\lambda_{i}$ are cardinals for every $i\in I$ . If $\kappa_{i}\leq\lambda_{i}$ for every $i\in I$ , then

	$\displaystyle\sum_{i\in I}\kappa_{i}$	$\displaystyle\leq\sum_{i\in I}\lambda_{i}.$
	$\displaystyle\prod_{i\in I}\kappa_{i}$	$\displaystyle\leq\prod_{i\in I}\lambda_{i}.$

If, moreover, $\kappa_{i}<\lambda_{i}$ for all $i\in I$ , then we have König’s theorem.

\displaystyle\sum_{i\in I}\kappa_{i}

\displaystyle<\,\prod_{i\in I}\lambda_{i}.

If $\kappa_{i}=\kappa$ for every $i$ in the index set $I$ , then

	$\displaystyle\sum_{i\in I}\kappa_{i}$	$\displaystyle=\kappa\|I\|.$
	$\displaystyle\prod_{i\in I}\kappa_{i}$	$\displaystyle=\kappa^{\|I\|}.$

Thus it is possible to define exponentiation in terms of multiplication, and multiplication in terms of addition.

Title	cardinal arithmetic
Canonical name	CardinalArithmetic
Date of creation	2013-03-22 14:15:13
Last modified on	2013-03-22 14:15:13
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	38
Author	yark (2760)
Entry type	Topic
Classification	msc 03E10
Related topic	OrdinalArithmetic
Related topic	CardinalNumber
Related topic	CardinalExponentiationUnderGCH
Related topic	CardinalityOfTheContinuum
Defines	cardinal addition
Defines	cardinal multiplication
Defines	cardinal exponentiation
Defines	sum of cardinals
Defines	product of cardinals
Defines	addition
Defines	multiplication
Defines	exponentiation
Defines	sum
Defines	product