Actual Infinity - Opposition From The Intuitionist School

Opposition From The Intuitionist School

The mathematical meaning of the term "actual" in actual infinity is synonymous with definite, completed, extended or existential, 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.

For intuitionists, infinity is described as potential; terms synonymous with this notion are becoming or constructive. For example, Stephen Kleene describes the notion of a Turing machine tape as "a linear tape, (potentially) infinite in both directions." To access memory on the tape, a Turing machine moves a read head along it in finitely many steps: the tape is therefore only "potentially" infinite, since while there is always the ability to take another step, infinity itself is never actually reached.

Mathematicians generally accept actual infinities. Georg Cantor is the most significant mathematician who defended actual infinities, equating the Absolute Infinite with God. He decided that it is possible for natural and real numbers to be definite sets, and that if one rejects the axiom of Euclidean finiteness (that states that actualities, singly and in aggregates, are necessarily finite), then one is not involved in any contradiction.

The philosophical problem of actual infinity concerns whether the notion is coherent and epistemically sound.