Constructive Objects

objects which can be investigated and operated on without involving abstractions of infinity stronger than the abstraction of potential realizability, which ignores practical limitations in the construction of concrete or abstract objects in space, time, or matter. If, for example, words formed from the letters of a particular alphabet are viewed as constructive objects, then this abstraction permits the investigation of words of any finite length. Again, in the case of the natural numbers it permits the investigation of integers of any finite magnitude, and so on.

As one of the fundamental concepts of modern mathematics, logic, and the theory of algorithms, the general concept of constructive objects is not defined but merely exemplified (as has been done above). At the same time, in particular constructive (logical-) mathematical theories we usually consider only constructive objects of a certain “standard” type, which, as a rule, are defined inductively. Then a general definition of the concept of constructive objects is superfluous. Words from letters of a certain fixed alphabet serve as such “standard” constructive objects. In A. A. Markov’s theory of normal algorithms (in some modifications, the theory of algorithms) such “standard” constructive objects are words in letters of a certain fixed alphabet, and in formalized arithmetic they are the natural numbers.