Actual infinity

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In the philosophy of mathematics, the abstraction of actual infinity, also called completed infinity,^[1] involves infinite entities as given, actual and completed objects.

Since Greek antiquity, the concept of actual infinity has been a subject of debate among philosophers. Also, the question of whether the Universe is infinite is still a debate between physicists.

The concept of actual infinity has been introduced in mathematics near the end of the 19th century by Georg Cantor, with his theory of infinite sets, later formalized into Zermelo–Fraenkel set theory. This theory, which is presently commonly accepted as a foundation of mathematics, contains the axiom of infinity, which means that the natural numbers form a set (necessarily infinite). A great discovery of Cantor is that, if one accept infinite sets, then there are different sizes (cardinalities) of infinite sets, and, in particular, the cardinal of the continuum of the real numbers is strictly larger than the cardinal of the natural numbers.

Actual infinity is to be contrasted with potential infinity, in which an endless process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps. This type of process occurs in mathematics, for instance, in standard formalizations of the notions of an infinite series, infinite product, or limit.^[2]