Actual Infinity - Aristotle's Potential–actual Distinction

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.

But Plato has two infinities, the Great and the Small. —Physics, book 3, chapter 4.

Aristotle also distinguished between actual and potential infinities. An actual infinity is something which is completed and definite and consists of infinitely many elements. According to Aristotle, the idea is paradoxical, both in theory and in nature. In respect to addition, a potentially infinite sequence or series would be endless: another element can always be counted for a sequence or series that is inexhaustible.

For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different. —Aristotle, Physics, book 3, chapter 6.

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, for example, the one that starts as 1, 0.5, 0.25, 0.125, 0.0625. According to Aristotle, the process of dividing a continuum need never come to an end. (In transfinitists, one starts with the notion that the limit value 0 exists and is reached).

For the fact that the process of dividing never comes to an end ensures that this activity exists potentially, but not that the infinite exists separately. — Metaphysics, book 9, chapter 6.

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 nothing to do with actualization in nature.