Counting is the process of determining the number of elements of a finite set of objects; that is, determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term enumeration refers to uniquely identifying the elements of a Finite set (combinatorial) set or infinite set by assigning a number to each element.
Counting sometimes involves numbers other than one; for example, when counting money, counting out change, "counting by twos" (2, 4, 6, 8, 10, 12, ...), or "counting by fives" (5, 10, 15, 20, 25, ...).
There is archaeological evidence suggesting that humans have been counting for at least 50,000 years. An Introduction to the History of Mathematics (6th Edition) by Howard Eves (1990) p.9 Counting was primarily used by ancient cultures to keep track of social and economic data such as the number of group members, prey animals, property, or debts (that is, accountancy). Notched bones were also found in the Border Caves in South Africa, which may suggest that the concept of counting was known to humans as far back as 44,000 BCE. The development of counting led to the development of mathematical notation, , and writing.
Counting a small set of objects, especially over time, can be accomplished efficiently with tally marks: making a mark for each number and then counting all of the marks when done tallying. Tallying is base 1 counting.
Finger counting is convenient and common for small numbers. Children count on fingers to facilitate tallying and for performing simple mathematical operations. Older finger counting methods used the four fingers and the three bones in each finger (phalanges) to count to twelve.
Other handgesture systems are also in use, for example the Chinese system by which one can count to 10 using only gestures of one hand. With finger binary it is possible to keep a finger count up to .Various devices can also be used to facilitate counting, such as and .
Similar counting is involved in East Asian age reckoning, in which are considered to be 1 at birth.
Musical terminology also uses inclusive counting of intervals between notes of the standard scale: going up one note is a second interval, going up two notes is a third interval, etc., and going up seven notes is an octave.
Many children at just 2 years of age have some skill in reciting the count list (that is, saying "one, two, three, ..."). They can also answer questions of ordinality for small numbers, for example, "What comes after three?". They can even be skilled at pointing to each object in a set and reciting the words one after another. This leads many parents and educators to the conclusion that the child knows how to use counting to determine the size of a set.Fuson, K.C. (1988). Children's counting and concepts of number. New York: Springer–Verlag. Research suggests that it takes about a year after learning these skills for a child to understand what they mean and why the procedures are performed.Le Corre, M., & Carey, S. (2007). One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition, 105, 395–438.Le Corre, M., Van de Walle, G., Brannon, E. M., Carey, S. (2006). Revisiting the competence/performance debate in the acquisition of the counting principles. Cognitive Psychology, 52(2), 130–169. In the meantime, children learn how to name cardinalities that they can subitize.
Many sets that arise in mathematics do not allow a bijection to be established with {1, 2, ..., n} for any natural number n; these are called , while those sets for which such a bijection does exist (for some n) are called . Infinite sets cannot be counted in the usual sense; for one thing, the mathematical theorems which underlie this usual sense for finite sets are false for infinite sets. Furthermore, different definitions of the concepts in terms of which these theorems are stated, while equivalent for finite sets, are inequivalent in the context of infinite sets.
The notion of counting may be extended to them in the sense of establishing (the existence of) a bijection with some wellunderstood set. For instance, if a set can be brought into bijection with the set of all natural numbers, then it is called "countably infinite." This kind of counting differs in a fundamental way from counting of finite sets, in that adding new elements to a set does not necessarily increase its size, because the possibility of a bijection with the original set is not excluded. For instance, the set of all (including negative numbers) can be brought into bijection with the set of natural numbers, and even seemingly much larger sets like that of all finite sequences of rational numbers are still (only) countably infinite. Nevertheless, there are sets, such as the set of , that can be shown to be "too large" to admit a bijection with the natural numbers, and these sets are called "uncountable set." Sets for which there exists a bijection between them are said to have the same cardinality, and in the most general sense counting a set can be taken to mean determining its cardinality. Beyond the cardinalities given by each of the natural numbers, there is an infinite hierarchy of infinite cardinalities, although only very few such cardinalities occur in ordinary mathematics (that is, outside set theory that explicitly studies possible cardinalities).
Counting, mostly of finite sets, has various applications in mathematics. One important principle is that if two sets X and Y have the same finite number of elements, and a function is known to be injective, then it is also surjective, and vice versa. A related fact is known as the pigeonhole principle, which states that if two sets X and Y have finite numbers of elements n and m with , then any map is not injective (so there exist two distinct elements of X that f sends to the same element of Y); this follows from the former principle, since if f were injective, then so would its restriction to a strict subset S of X with m elements, which restriction would then be surjective, contradicting the fact that for x in X outside S, f( x) cannot be in the image of the restriction. Similar counting arguments can prove the existence of certain objects without explicitly providing an example. In the case of infinite sets this can even apply in situations where it is impossible to give an example.
The domain of enumerative combinatorics deals with computing the number of elements of finite sets, without actually counting them; the latter usually being impossible because infinite families of finite sets are considered at once, such as the set of of {1, 2, ..., n} for any natural number n.

