In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite sets.
Cardinality is defined in terms of bijective functions. Two sets have the same cardinal number if and only if there is a bijection between them. In the case of finite sets, this agrees with the intuitive notion of size. In the case of infinite sets, the behavior is more complex. A fundamental theorem due to Georg Cantor shows that it is possible for infinite sets to have different cardinalities, and in particular the set of real numbers and the set of natural numbers do not have the same cardinal number. It is also possible for a proper subset of an infinite set to have the same cardinality as the original set, something that cannot happen with proper subsets of finite sets.
There is a transfinite sequence of cardinal numbers:
This sequence starts with the natural numbers (finite cardinals), which are followed by the aleph numbers (infinite cardinals of well-ordered sets). The aleph numbers are indexed by ordinal numbers. Under the assumption of the axiom of choice, this transfinite sequence includes every cardinal number. If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs.
Cardinality is studied for its own sake as part of set theory. It is also a tool used in branches of mathematics including combinatorics, abstract algebra, and mathematical analysis.
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