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cardinal arithmetic

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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.

$\displaystyle \kappa+\lambda$	$\displaystyle =\vert A\cup B\vert.$
$\displaystyle \kappa\lambda$	$\displaystyle =\vert A\times B\vert.$
$\displaystyle \kappa^\lambda$	$\displaystyle =\vert A^B\vert.$

(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 $\prod_{i\in I}B_i$ , 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:

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

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

$\displaystyle \kappa+\lambda$	$\displaystyle =\max(\kappa,\lambda).$
$\displaystyle \kappa\lambda$	$\displaystyle =\max(\kappa,\lambda),$ provided $\displaystyle \kappa\ne0\ne\lambda.$

Also notable is that if $\kappa$ and $\lambda$ are cardinals with $\lambda$ infinite and $2 \le \kappa \le 2^\lambda$ , 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 $\mu\le\kappa$ and $\nu\le\lambda$ , then

$\displaystyle \mu+\nu$	$\displaystyle \le\kappa+\lambda.$
$\displaystyle \mu\nu$	$\displaystyle \le\kappa\lambda.$
$\displaystyle \mu^\nu$	$\displaystyle \le\kappa^\lambda,$ unless $\displaystyle \mu=\nu=\kappa=0<\lambda.$

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 $\kappa_i\le\lambda_i$ for every $i\in I$ , then

$\displaystyle \sum_{i\in I}\kappa_i$	$\displaystyle \le\sum_{i\in I}\lambda_i.$
$\displaystyle \prod_{i\in I}\kappa_i$	$\displaystyle \le\prod_{i\in I}\lambda_i.$

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

$\displaystyle \sum_{i\in I}\kappa_i$	$\displaystyle =\kappa\vert I\vert.$
$\displaystyle \prod_{i\in I}\kappa_i$	$\displaystyle =\kappa^{\vert I\vert}.$

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

"cardinal arithmetic" is owned by yark.

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Also defines:

cardinal addition, cardinal multiplication, cardinal exponentiation, sum of cardinals, product of cardinals, addition, multiplication, exponentiation, sum, product

This object's parent.

Attachments:: König's theorem (Theorem) by yark
cardinal exponentiation under GCH (Theorem) by yark
idempotency of infinite cardinals (Definition) by CWoo

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Cross-references: König's theorem, Cantor's theorem, infinite, real, pairwise disjoint, union, index set, numbers, well-defined, operations, functions, cardinality, disjoint, cardinal numbers
There are 21 references to this entry.

This is version 35 of cardinal arithmetic, born on 2004-03-13, modified 2007-07-06.
Object id is 5701, canonical name is CardinalArithmetic.
Accessed 16094 times total.

Classification:

AMS MSC:

03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)

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