Abstraction of actual infinity

A mathematical idealization, related to a certain form of the idea of infinity in mathematics — the idea of the so-called actual infinity.

As applied to constructive processes of potentially infinite duration (e.g. the generation of successive positive integers starting from zero), abstraction of actual infinity consists of ignoring the fact that such processes do not terminate in principle and in considering the results of such processes on the assumption that they have terminated, viz. that the sets of objects have been generated. The resulting sets (objects) are then mentally regarded as actual "finished" objects. The application of the abstraction of actual infinity to the above example makes it possible to consider the set of all non-negative integers — the natural sequence — as a mathematical object.

Logically, acceptance of the abstraction of actual infinity leads to the acceptance of the law of the excluded middle as a logical principle.

Abstraction of actual infinity is particularly important in the construction of mathematics on the base of the general theory of sets, established by G. Cantor. Abstraction of actual infinity, being a far-reaching idealization, especially so when repeatedly employed in conjunction with other idealizations, generates objects whose "tangibility" is indirect, as a result of which the problems involved in understanding propositions concerning such objects meet with certain difficulties. Unrestricted use of abstraction of actual infinity in mathematics as a legitimate method of generating mathematical objects met with objections by a number of mathematicians (L. Kronecker, C.F. Gauss, D. Hilbert, H. Weyl, and others). Positive programs of mathematical construction based on abstraction of potential realizability, without recourse to abstraction of actual infinity, were proposed by L.E.J. Brouwer (see Intuitionism) and by A.A. Markov (cf. Constructive mathematics).