Actual infinity

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Actual infinity is the notion that all (natural, real etc.) numbers can be enumerated in any sense sufficiently definite for them to form a set together. Hence, in the philosophy of mathematics, the abstraction of actual infinity is the acceptance of infinite entities, such as the set of all natural numbers or an arbitrary sequence of rational numbers, as given objects.

The mathematical meaning of the term actual in actual infinity is synonymous with definite, not to be mistaken for physically existing. The question of whether natural or real numbers form definite sets is therefore independent of the question of whether infinite things exist actually in nature.

To reject the abstraction of actual infinity, as in intuitionism (see also intuitionistic logic), requires the reconstruction of the foundations of set theory and calculus as constructivist set theory and constructivist analysis respectively. One mathematician who accepted actual infinities was Georg Cantor, who decided that it is possible for natural and real numbers to be definite sets, and that if we reject the axiom of Euclidean finiteness (that states that actualities, singly and in aggregates, are necessarily finite), then we are not involved in any contradiction. This rejection was also adopted by philosophers and mathematicians in the 20th century, such as Michael Dummett.

The mathematical problem of actual infinities is whether this rejection is justified or not.