Constructive object

A name used for mathematical objects that arise as a result of completing so-called constructive processes. For the description of some constructive process or other "… it is usually assumed that the objects which figure in a given consideration as indecomposable initial objects are clearly characterized; it is assumed that a list of the rules for forming new objects from already constructed objects that figure in the given consideration as descriptions of allowable steps of constructive processes is given; it is assumed that the formation processes are realized by separate steps, where the choice of a new step is carried out under the restrictions determined by the already constructed objects and by the collection of those formation rules that actually can be applied to the already constructed objects" (cf. [4]). This description of a constructive process, and thus of a constructive object, cannot, of course, have the pretension of being an exact mathematical definition. However, concrete mathematical theories always deal only with those concrete types of constructive objects that have an exact characterization. The description of a constructive object given above may serve in those situations as a means for choosing corresponding exact definitions.

Words over a fixed alphabet provide an example of an exactly defined type of constructive objects (the letters in this alphabet play the role of initial objects; new words are obtained from already constructed ones by the addition on the right (cf. [3], Sect. 17) of letters of the alphabet considered). Other examples of types of constructive objects are finite graphs (cf. Graph), finite abstract topological complexes (cf. Complex), and relay-contact schemes (the choice of the corresponding initial objects and formation rules present no difficulties). Also, rational numbers, algebraic polynomials, algorithms, and calculi of various well-defined types (cf. Algorithm; Calculus), finitely-presented groups and other analogous mathematical objects may be defined as constructive objects.

Constructive objects play an important role in those mathematical theories in which the need arises to consider objects that allow for a clear individual specification by means of some mathematical symbolism. Within set-theoretical mathematics, with its unlimited use of the abstraction of actual infinity, constructive objects and arbitrary sets of constructive objects are considered along with, and on the same level as, simple mathematical objects, among which the constructive objects are distinguished only by their higher "tangibility" . Within constructive mathematics, constructive objects (or objects determined by them) form the naturally admissible type of mathematical objects for consideration. Their consideration here is based on a rejection of the abstraction of actual infinity as well as on a special constructive logic, taking the features of the definition of constructive objects into account. See also Constructive mathematics.

References

[1]	A.A. Markov, "On constructive mathematics" Amer. Math. Soc. Translations (2) , 98 (1971) pp. 1–9 Trudy Mat. Inst. Steklov. , 67 (1962) pp. 8–14
[2]	A.A. Markov, "On the logic of constructive mathematics" Vestnik Moskov. Univ. Ser. I Mat. Mekh. , 25 (1970) pp. 7–29 (In Russian)
[3]	A.A. Markov, N.M. [N.M. Nagornyi] Nagorny, "The theory of algorithms" , Reidel (1988) (Translated from Russian) Zbl 0663.03023
[4]	N.A. Shanin, "Constructive real numbers and constructive function spaces" , Transl. Math. Monogr. , 21 , Amer. Math. Soc. (1968) (Translated from Russian)