Definition:Cardinality/Infinite
Definition
Let $S$ be an infinite set.
The cardinality $\card S$ of $S$ can be indicated as:
- $\card S = \infty$
However, it needs to be noted that this just means that the cardinality of $S$ cannot be assigned a number $n \in \N$.
It means that $\card S$ is at least $\aleph_0$ (aleph null).
Also defined as
Some authors, working to a particular mathematical agenda, do not discuss the cardinality of an infinite set, and instead limit their definition of this concept to the finite case.
Some others gloss over the definition for the cardinality of a finite set, perhaps on the understanding that the definition is trivial, and instead raise the concept only in the infinite case.
Also known as
Some authors prefer the term order instead of cardinality, particularly in the context of finite sets.
Georg Cantor used the term power and equated it with the term cardinal number, using the notation $\overline {\overline M}$ for the cardinality of $M$.
Some sources cut through all the complicated language and call it the size.
Some sources use $\map {\#} S$ (or a variant) to denote set cardinality. This notation has its advantages in certain contexts, and is used on occasion on this website.
Others use $\map C S$, but this is easy to confuse with other uses of the same or similar notation.
A clear but relatively verbose variant is $\Card \paren S$ or $\operatorname{card} \paren S$.
1968: A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis use $\map m A$ for the power of the set $A$.
Further notations are $\map n A$ and $\overline A$.
Also see
- Definition:Aleph Number ($\aleph_0, \aleph_1, \ldots$)
- Definition:Beth Number ($\beth_0, \beth_1, \ldots$)
Sources
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 2.5$: The power of a set
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Mappings: $\S 15$
- 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text{A}.6$: Cardinality