Set

An aggregate, totality, collection of any objects whatever, called its elements, which have a common characteristic property. "A set is many, conceivable to us as one" (G. Cantor). This is not in a true sense a logical definition of the notion of a set, rather it is just an explanation (because defining the notion means finding a generic idea to which the given idea belongs as a species; but a set is, unfortunately, itself a broad notion in mathematics and logic). In this connection it is possible either to give a list of the elements of the set, an enumeration of it, or to give a rule for determining whether or not a given object belongs to the set considered, a description of it (moreover, the first is really acceptable only when the question concerns finite sets).

For a meaningful development of "naive" set theory such an explanation is quite sufficient, because for the mathematical theory it is only essential to define the relations between the elements of a set (or between the sets themselves), and not their nature. To describe sets which may be elements of another set, to avoid the so-called antinomies (cf. Antinomy) one introduces, for example, the terminology "class" . And then, speaking more formally, set theory deals with objects called classes (cf. Class), for which there is defined a relation of membership, and a set itself is defined as a class which is an element of some class.

Recently the unifying role of category theory (and, in particular, the notion of a universal set) has become more clear. The construction of a category is based on axiomatic set theory and allows one to consider, for example, such "large" collections as the category of all sets, groups, topological spaces, etc.

References

 [1] , Lectures on the sets of Georg Cantor , New ideas in mathematics , 6 , St. Petersburg (1914) (In Russian) [2] Yu.A. Shikhanovich, "Introduction to modern mathematics" , Moscow (1965) (In Russian) [3] N.I. Kondakov, "Logical reference-dictionary" , Moscow (1975) (In Russian) [4] N. Bourbaki, "Elements of mathematics. Theory of sets" , Addison-Wesley (1968) (Translated from French) [5] P.S. Novikov, "Elements of mathematical logic" , Oliver & Boyd and Acad. Press (1964) (Translated from Russian) [6] P.M. Cohn, "Universal algebra" , Reidel (1981) [7] J.R. Shoenfield, "Mathematical logic" , Addison-Wesley (1967)

Cf. also Universe; Descriptive set theory.

References

 [a1] G. Grätzer, "Universal algebra" , Springer (1979) [a2] P.R. Halmos, "Naive set theory" , v. Nostrand-Reinhold (1960) [a3] H. Meschkowski, "Hundert Jahre Mengenlehre" , DTV (1973) [a4] P. Suppes, "Axiomatic set theory" , v. Nostrand (1965) [a5] A. Levy, "Basic set theory" , Springer (1979) [a6] A.A. Fraenkel, "Abstract set theory" , North-Holland (1961) [a7] J.R. Shoenfield, "Axioms of set theory" J. Barwise (ed.) , Handbook of mathematical logic , North-Holland (1978) pp. 321–345 [a8] J. Barwise (ed.) , Handbook of mathematical logic , North-Holland (1977) [a9] G. Cantor, "Contributions to the founding of the theory of transfinite numbers" , Dover, reprint (1955) (Translated from German)
How to Cite This Entry:
Set. M.I. Voitsekhovskii (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Set&oldid=11955
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098