latter is incapable of increase and is therefore indeterminable as a mathematical concept. This mistake we find, for example, in Pantheism. (G. Cantor, Über verschiedene Standpunkte in bezug auf das aktuelle Unendliche, in Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, pp. 375, 378) Current mathematical practice Actual infinity is now commonly accepted, because mathematicians have learned how to construct algebraic statements using it. For example, one may write down a symbol, , with the verbal description that " stands for completed (countable) infinity". This symbol may be added as an ur-element to any set. One may also provide axioms that define addition, multiplication and inequality; specifically, ordinal arithmetic, such that expressions like can be interpreted as "any natural number is less than completed infinity". Even "common sense" statements such as are possible and consistent. The theory is sufficiently well developed, that rather complex algebraic expressions, such as , and even can be interpreted as valid algebraic expressions, can be given a verbal description, and can be used in a wide variety of theorems and claims in a consistent and meaningful fashion. The ability to define ordinal numbers in a consistent, meaningful way, renders much of the debate moot; whatever personal opinion one may hold about infinity or constructability, the existence of a rich theory for working with infinities using the tools of algebra and logic is clearly in hand. 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. 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"