The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

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 induction. 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.


Comments (2)

Comment #3568 by Christian Hildebrandt on

Is addition of cardinals as uninteresting as multiplication since, if or are infinite, ?

Comment #3692 by on

Yes, of course. OK, I think you are telling me to remove the term interesting which I did. See changes here.


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