Abstraction of potential realizability

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A mathematical idealization, related to a certain form of the concept of infinity in mathematics — the idea of a potential infinity.

As applied to constructive processes which can, in principle, be indefinitely extended (e.g. the successive generation of positive integers starting from zero), the abstraction of potential realizability consists in ignoring any possible spatial, temporal or material obstacles to the realization of each successive step of the process, and to consider each step as potentially realizable. The application of the abstraction of potential realizability to the example given above is tantamount to assuming that a unit can be added to any natural number, that it is possible to form the sum of any two natural numbers, etc., but it does not imply the idea of the existence of the natural sequence as an actual "infinite object" .

The acceptance of the abstraction of potential realizability logically leads to the principle of mathematical induction.

Abstraction of potential realizability plays a special role in constructive mathematics, in which propositions concerning the existence of constructive objects that satisfy given conditions are regarded as propositions on the potential realizability of such objects.

See also Abstraction, mathematical.

How to Cite This Entry:
Abstraction of potential realizability. >N.M. Nagornyi (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098