In set theory, counting is the act of placing things in a one-to-one correspondence with a subset of the natural numbers (not necessarily a proper subset) in such a way that the numbers are used in order with no gaps (each subsequent number is exactly 1 greater than the previous).
If a collection or set of things can be so counted, it is called countable. The "number of things" in a set is called its size or cardinality. The cardinality of countable sets can be finite or countably infinite. Sets which cannot be counted (uncountable sets) include those with cardinality greater than aleph null, the cardinality of the natural numbers (see Transfinite number). See also Set theory for a review of set terminology.
- 1, 2, 3, 4, 5, ...
and this is how we begin to count things.
More complicated patterns of counting are studied in elementary combinatorics.
In some locations, the set of counting numbers starts with 0. As such, positive integer may be a perferred term to use instead.