Actual infinity

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Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given objects.

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Aristotle's Potential-Actual Distinction

Aristotle handled the topic of infinity in Physics and in Metaphysics. Aristotle distinguished between infinity in respect to addition and in respect to division.

Aristotle also distinguished between actual and potential infinities. An actual infinity is something which is completed and definite and consists of infinitely many elements, and according to Aristotle, a paradoxical idea, both in theory and in nature. In respect to addition, a potentially infinite sequence or a series is potentially endless; being a potentially endless series means that one element can always be added to the series after another, and this process of adding elements is never exhausted.

As an example of a potentially infinite series in respect to increase, one number can always be added after another in the series that starts 1,2,3,... but the process of adding more and more numbers cannot be exhausted or completed, because there is no end to the process.

In respect to division, a potentially infinite series of divisions is e.g. the one that starts as 1, 0.5, 0.25, 0.125, 0.0625. According to Aristotle, the process of division never comes to an end, and the limit value 0 is never reached, although the division can be continued as long as one wants. This is a crucial difference to the transfinitists, who start with the very notion that the limit value exists and is reached (this is not to say that 0 would not exist; zero is at our disposal).

In contrast to the potential infinity, all the elements of an actually infinite (= transfinite) set are assumed to exist together simultaneously as a completed totality. The term 'transfinite' ought to be used instead of 'actually infinite' to denote the transfinite sets, because the set-theoretical notion of actual infinity has got nothing to do with actualization in nature.

Opposition from the Intuitionist school

The mathematical meaning of the term "actual" in actual infinity is synonymous with definite, completed, extended or existential[1], but 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 physically in nature.

Proponents of intuitionism, from Kronecker onwards, reject the claim that there are actually infinite mathematical objects or sets. (Also, according to Aristotle, a completed infinity cannot exist even as an idea in the mind of a human.) Consequently, they reconstruct the foundations of mathematics in a way that does not assume the existence of actual infinities. On the other hand, constructive analysis does accept the existence of the completed infinity of the integers.

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