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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|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|Antinomy]]) one introduces, for example, the terminology  "class" . And then, speaking more formally, set theory deals with objects called classes (cf. [[Class|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|category]] theory (and, in particular, the notion of a [[Universal set|universal set]]) has become more clear. The construction of a category is based on [[Axiomatic set theory|axiomatic set theory]] and allows one to consider, for example, such  "large"  collections as the category of all sets, groups, topological spaces, etc.

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



====Comments====
Cf. also [[Universe|Universe]]; [[Descriptive set theory|Descriptive set theory]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Grätzer,   "Universal algebra" , Springer  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Halmos,   "Naive set theory" , v. Nostrand-Reinhold  (1960)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Meschkowski,   "Hundert Jahre Mengenlehre" , DTV  (1973)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Suppes,   "Axiomatic set theory" , v. Nostrand  (1965)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Levy,   "Basic set theory" , Springer  (1979)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.A. Fraenkel,   "Abstract set theory" , North-Holland  (1961)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.R. Shoenfield,   "Axioms of set theory"  J. Barwise (ed.) , ''Handbook of mathematical logic'' , North-Holland  (1978)  pp. 321–345</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J. Barwise (ed.) , ''Handbook of mathematical logic'' , North-Holland  (1977)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  G. Cantor,   "Contributions to the founding of the theory of transfinite numbers" , Dover, reprint  (1955)  (Translated from German)</TD></TR></table>