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