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.

Comments (4)

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.

Comment #4582 by Jessica on

Shouldn't the comment about addition and multiplication specify that at least one of and should be infinite? You don't need both cardinals to be infinite.

Comment #4762 by on

Yes, Jessica, you are correct (although you need to make sure neither is for the thing about products to work). But it is still correct as it stands and we don't ever explicitly refer to this, so I'm going to leave it as is.

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 000D. Beware of the difference between the letter 'O' and the digit '0'.