Cardinal

Idea

In set theory, cardinality is a way of classifying sets, by the kinds of maps possible between any two sets by ‘size’.

Definition

The cardinal of a set $X$ is the equivalence class, written $|X|$ of sets that have bijections between $X$ and other sets. Let $X$, $Y$ be sets, then the following are considered definitions:

$$\begin{array}{rl}|X|=|Y|& ⟺\exists f:X\to Y.\text{f is a bijection}\\ |X|\le |Y|& ⟺\exists f:X\to Y.\text{f is a injection}\\ |X|<|Y|& ⟺|X|\le |Y|\wedge |X|\ne |Y|\end{array}$$

Finite Cardinality

The cardinality is written as $|X|=n$ for $n\in \mathbb{N}$ iff $|X|=|[n]|$ where $[n]$ is the canonical finite set $\{m\in \mathbb{N}.m<n\}$. The first infinite cardinal is denoted ${\aleph }_0$. Each subsequent cardinal is written ${\aleph }_{\alpha }$ for any other ordinal $\alpha$.

Remarks

• Cardinals are in general proper classes. An alternative definition defines then as initial ordinals.

Common Facts

• Cantor’s theorem: For any set $X$, the cardinality of its power set is strictly greater than the cardinality of $X$. Symbolically, $\kappa <2^{\kappa }$.
• Schröder-Bernstein theorem: If there exist injective functions from $A$ to $B$ and from $B$ to $A$, then there exists a bijective function between $A$ and $B$.
• Infinite arithmetic: For infinite cardinals $\kappa$ and $\lambda$, assuming the Axiom of Choice, addition and multiplication simplify to $\kappa +\lambda =\kappa \cdot \lambda =\max (\kappa ,\lambda )$.
• [^1]Aleph numbers: The infinite well-ordered cardinals are denoted by the Hebrew letter aleph indexed by ordinals: ${\aleph }_0,{\aleph }_1,{\aleph }_2,\dots$. The cardinality of the natural numbers is ${\aleph }_0$.
• The continuum: The cardinality of the real numbers, denoted $\mathfrak{c}$, equals $2^{{\aleph }_0}$. The Continuum hypothesis states that $\mathfrak{c}={\aleph }_1$, which is independent of ZFC.

Relation to the Skeleton of Set

In category theory, a skeleton of a category is a full subcategory where no two distinct objects are isomorphic, and every object in the original category is isomorphic to an object in the subcategory.

In the category of sets and functions, denoted Set, isomorphisms are bijections. Two sets are isomorphic if and only if they have the same cardinality.

A skeleton of Set therefore contains exactly one set of each cardinality. The class of cardinal numbers, when defined as initial ordinals via the Von Neumann cardinal assignment, provides a canonical choice for the objects of this skeleton. We view the skeleton of Set as the category whose objects are the cardinal numbers and whose morphisms are arbitrary functions between them.

References

• jech2003-set-theory
• maclane1978-categories