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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.

Elementary notions

The set of positive natural numbers (originally the only "numbers" children know of) begins:

1, 2, 3, 4, 5, ...

and this is how we begin to count things.

In arithmetic, children learn that counting is a special case of addition in which 1 is added repeatedly. That is, , , , etc.

Repeated addition of the same number can be represented by multiplication, and repeated multiplication by exponentiation ("powers"). See Operations of arithmetic for details.

More complicated patterns of counting are studied in elementary combinatorics.

Regional differences

In some locations, the set of counting numbers starts with 0. As such, positive integer may be a perferred term to use instead.