This theory, which is presently commonly accepted as a foundation of mathematics, contains the axiom of infinity, which means that the natural numbers form a set (necessarily infinite).
Actual infinity is to be contrasted with potential infinity, in which an endless process (such as "add 1 to the previous number") produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps.
Anaximander (610–546 BC) held that the apeiron was the principle or main element composing all things.
However, he said, mathematics relating to infinity was not deprived of its applicability by this impossibility, because mathematicians did not need the infinite for their theorems, just a finite, arbitrarily large magnitude.
[8] The overwhelming majority of scholastic philosophers adhered to the motto Infinitum actu non datur.
It is well known that in the Middle Ages all scholastic philosophers advocate Aristotle's "infinitum actu non datur" as an irrefutable principle.
(J. Baconthorpe [9, p. 96])During the Renaissance and by early modern times the voices in favor of actual infinity were rather rare.
Leibniz [9, p. 97])However, the majority of pre-modern thinkers[citation needed] agreed with the well-known quote of Gauss: I protest against the use of infinite magnitude as something completed, which is never permissible in mathematics.
Infinity is merely a way of speaking, the true meaning being a limit which certain ratios approach indefinitely close, while others are permitted to increase without restriction.
Gauss [in a letter to Schumacher, 12 July 1831])Actual infinity is now commonly accepted in mathematics, although the term is no longer in use, being replaced by the concept of infinite sets.
This drastic change was initialized by Bolzano and Cantor in the 19th century, and was one of the origins of the foundational crisis of mathematics.
(G. Cantor, Über verschiedene Standpunkte in bezug auf das aktuelle Unendliche, in Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, pp.
Proponents of intuitionism, from Kronecker onwards, reject the claim that there are actually infinite mathematical objects or sets.
[13] For example, Stephen Kleene describes the notion of a Turing machine tape as "a linear 'tape', (potentially) infinite in both directions.
The present-day conventional finitist interpretation of ordinal and cardinal numbers is that they consist of a collection of special symbols, and an associated formal language, within which statements may be made.
The philosophical problem of actual infinity concerns whether the notion is coherent and epistemically sound.