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# 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 $\prod_{{i\in I}}B_{i}$, 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.

## Mathematics Subject Classification

03E10*no label found*

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