The Stacks project

3.6 Cardinality

The cardinality of a set $A$ is the least ordinal $\alpha $ such that there exists a bijection between $A$ and $\alpha $. We sometimes use the notation $\alpha = |A|$ to indicate this. We say an ordinal $\alpha $ is a cardinal if and only if it occurs as the cardinality of some set $A$—in other words, if $\alpha = |A|$. We use the greek letters $\kappa $, $\lambda $ for cardinals. The first infinite cardinal is $\omega $, and in this context it is denoted $\aleph _0$. A set is countable if its cardinality is $\leq \aleph _0$. If $\alpha $ is an ordinal, then we denote $\alpha ^+$ the least cardinal $> \alpha $. You can use this to define $\aleph _1 = \aleph _0^+$, $\aleph _2 = \aleph _1^+$, etc, and in fact you can define $\aleph _\alpha $ for any ordinal $\alpha $ by transfinite recursion. We note the equality $\aleph _1 = \omega _1$.

The addition of cardinals $\kappa , \lambda $ is denoted $\kappa \oplus \lambda $; it is the cardinality of $\kappa \amalg \lambda $. The multiplication of cardinals $\kappa , \lambda $ is denoted $\kappa \otimes \lambda $; it is the cardinality of $\kappa \times \lambda $. If $\kappa $ and $\lambda $ are infinite cardinals, then $\kappa \oplus \lambda = \kappa \otimes \lambda = \max (\kappa , \lambda )$. The exponentiation of cardinals $\kappa , \lambda $ is denoted $\kappa ^\lambda $; it is the cardinality of the set of (set) maps from $\lambda $ to $\kappa $. Given any set $K$ of cardinals, the supremum of $K$ is $\sup _{\kappa \in K} \kappa = \bigcup _{\kappa \in K} \kappa $, which is also a cardinal.