Editing Actual infinity (section)

== Opposition from the Intuitionist school ==
The mathematical meaning of the term "actual" in '''actual infinity''' is synonymous with '''definite''', '''completed''', '''extended''' or '''existential''',<ref name="Kleene 1952/1971:48.">Kleene 1952/1971:48.</ref> but not to be mistaken for ''physically existing''. The question of whether [[Natural number|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 [[Leopold Kronecker|Kronecker]] onwards, reject the claim that there are actually infinite mathematical objects or sets. 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''.<ref name="Kleene 1952/1971:48."/> For example, [[Stephen Kleene]] describes the notion of a [[Turing machine]] tape as "a linear 'tape', (potentially) infinite in both directions."<ref>Kleene 1952/1971:48 p. 357; also "the machine ... is supplied with a tape having a (potentially) infinite printing ..." (p. 363).</ref> 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.<ref>Or, the "tape" may be fixed and the reading "head" may move. Roger Penrose suggests this because: "For my own part, I feel a little uncomfortable about having our finite device moving a potentially infinite tape backwards and forwards. No matter how lightweight its material, an ''infinite'' tape might be hard to shift!" Penrose's drawing shows a fixed tape head labelled "TM" reading limp tape from boxes extending to the visual vanishing point. (Cf page 36 in Roger Penrose, 1989, ''The Emperor's New Mind'', Oxford University Press, Oxford UK, {{ISBN|0-19-851973-7}}). Other authors{{Who?|date=December 2022}} solve this problem by tacking on more tape when the machine is about to run out.</ref>

Mathematicians generally accept actual infinities.<ref>Actual infinity follows from, for example, the acceptance of the notion of the integers as a set, see J J O'Connor and E F Robertson, [https://mathshistory.st-andrews.ac.uk/HistTopics/Infinity/ "Infinity"].</ref> [[Georg Cantor]] is the most significant mathematician who defended actual infinities. 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 present-day conventional finitist interpretation of [[ordinal number|ordinal]] and [[cardinal number]]s is that they consist of a collection of special symbols, and an associated [[formal language]], within which statements may be made. All such statements are necessarily finite in length. The soundness of the manipulations is founded only on the basic principles of a formal language: [[term algebra]]s, [[term rewriting]], and so on. More abstractly, both (finite) [[model theory]] and [[proof theory]] offer the needed tools to work with infinities. One does not have to "believe" in infinity in order to write down algebraically valid expressions employing symbols for infinity.