SET THEORY

branch of mathematics that deals with the properties of well-defined collections of objects, which may be of a mathematical nature, such as numbers or functions, or not. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts. Sets may be finite or infinite. A finite set has a definite number of members; such a set might consist of all the integers from 1 to 1,000 or all marked bus stops along a given route. An infinite set has an endless number of members; all the positive integers or all points along a given line compose infinite sets. A set is commonly represented by a list of its members, or elements, enclosed within braces; the statement that a set called A comprises the numbers 1, 2, and 3 is made by the expression A = {1,2,3}. A set that has no members is called an empty set (or a null or void set) and is denoted by the symbol . Operations on sets include union, symbolized , and intersection, symbolized . The union of two sets is itself the set containing all members of the two original sets; the intersection includes only the members common to both original sets. Thus, if A = {1,2,3} and B = {3,4,5}, then A B = {1,2,3,4,5} and A B = {3}. Various relations can exist between sets. Equivalent sets have the same number of members; thus, if A = { j,k,l,m} and B = {1,2,3,4}, then A ~ B (A is equivalent to B). Equal sets, on the other hand, consist of the same members; if A = {o,p,q} and B = {p,q,o}, then A = B (A is equal to B). One set is a subset of another (symbolized ) if all the members of the first are members of the second; if A = {1,2,3} and B = {1,2,3,4,5}, then A B. Disjoint sets have no members in common. The complement of a set A with respect to a larger set B is the set A, which contains all members of B that are not members of A. Thus, if B is the set of all positive integers and A is the set of all even positive integers, then A is the set of all odd positive integers. The value of set theory is its usefulness in analyzing difficult concepts in mathematics and logic. The intuitive notion of a set, however, is probably even older than that of number: members of a herd of animals, for example, could be paired with stones in a sack or notches in a stick without engaging in the process of actually counting anything. As mathematics evolved, its practitioners used the idea of sets, but only informally until the 19th century, when George Boole and Georg Cantor discovered the value of clearly formulated sets in the analysis of problems in symbolic logic and number theory. Cantor, now regarded as the father of set theory, developed the general theory of transfinite numbers. He revolutionized the philosophical foundations of mathematics by boldly insisting on the actual infinite, that is, on the existence of infinite sets as mathematical objects on a par with numbers and finite sets. In erecting his theory, Cantor did not promulgate specific axioms as its basis, though his reasoning was based on certain assumptions. The first axioms explicitly formulated for set theory led to logical inconsistencies, as pointed out by Bertrand Russell and others around 1900. Resolution of the deficiencies has led to two alternative systems as bases of modern set theory. The first of these was initiated by Ernst Zermelo in 1908 and strengthened by Abraham Fraenkel and Thoralf Skolem in the early 1920s. The second was devised by John von Neumann in the late 1920s and later modified by Paul Bernays and Kurt Gdel. Both systems obviate the paradoxes that plagued earlier theories of sets, but neither has reached its proponents' goal of providing a complete basis for all of mathematics. branch of mathematics that deals with the properties of well-defined collections of objects, which may be of a mathematical nature, such as numbers or functions, or not. The theory is less valuable in direct application to ordinary experience than as a basis for precise and adaptable terminology for the definition of complex and sophisticated mathematical concepts. Between the years 1874 and 1897, the German mathematician and logician Georg Cantor created a theory of abstract sets of entities and made it into a mathematical discipline. This theory grew out of his investigations of certain concrete problems regarding certain types of infinite sets of real numbers (see analysis: Real analysis: The basic number systems). A set, wrote Cantor, is a collection of definite, distinguishable objects of perception or thought conceived as a whole. The objects are called elements or members of the set. The theory had the revolutionary aspect of treating infinite sets as mathematical objects that are on an equal footing with those that can be constructed in a finite number of steps. Since antiquity, a majority of mathematicians had carefully avoided the introduction into their arguments of the actual infinite (i.e., of sets containing an infinity of objects conceived as existing simultaneously, at least in thought). Since this attitude persisted until almost the end of the 19th century, Cantor's work was the subject of much criticism to the effect that it dealt with fictions; indeed, that it encroached on the domain of philosophers and violated the principles of religion. Once applications to analysis began to be found, however, attitudes began to change, and in the 1890s Cantor's ideas and results were gaining acceptance. By 1900, set theory was recognized as a distinct branch of mathematics. At just that time, however, it received a severe setback through the derivation of several contradictions in its superstructure (see below Cardinality and transfinite numbers). The main thrust of this article is to present an account of one response to such contradictions as these. The purpose of the development here related has been to provide an axiomatic basis for the theory of sets analogous to that developed for elementary geometry. The degree of success that has been achieved in this development, as well as the present stature of set theory, has been well expressed in the Bourbaki lments de mathmatique: Nowadays it is known to be possible, logically speaking, to derive practically the whole of known mathematics from a single source, The Theory of Sets. Robert R. Stoll The Editors of the Encyclopdia Britannica Additional reading Overviews are provided in Nicolas Bourbaki, Elements of Mathematics, vol. 1, Theory of Sets (1968, reissued 1974; originally published in French, 3rd ed. rev., 1966); Azriel Lvy, Basic Set Theory (1979), an explanation for the advanced student; and Herbert B. Enderton, Elements of Set Theory (1972), at the advanced undergraduate or beginning graduate level. Georg Cantor, Contributions to the Founding of the Theory of Transfinite Numbers, trans. from German (1915, reissued 1955), is of historical interest. I. Grattan-Guinness (ed.), From the Calculus to Set Theory, 16301910 (1980), is a historical introduction.Various aspects of set theory are addressed by William S. Hatcher, Foundations of Mathematics (1968), an overall view of axiomatic set theory and its relationship to the foundations of mathematics; Patrick Suppes, Axiomatic Set Theory (1960, reprinted 1972); Gaisi Takeuti and Wilson M. Zaring, Introduction to Axiomatic Set Theory, 2nd ed. (1982); Michael D. Potter, Sets: An Introduction (1990), an axiomatic development of set theory; Kurt Gdel, The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory (1940, reissued 1970); Paul J. Cohen, Set Theory and the Continuum Hypothesis (1966), proofs of the independence of AC and CH; Herman Rubin and Jean E. Rubin, Equivalents of the Axiom of Choice, II (1985); Joseph R. Shoenfield, Mathematical Logic (1967), a development of ZF and the independence proofs; Robert R. Stoll, Set Theory and Logic (1963, reissued 1979), an informal development of ZF; Gregory H. Moore, Zermelo's Axiom of Choice: Its Origins, Development, and Influence (1982), a good account of an important topic; Elliott Mendelson, Introduction to Mathematical Logic, 3rd ed. (1987), a formal development of NBG; Yiannis N. Moschovakis, Descriptive Set Theory (1980), an extensive treatment; A.S. Kechris and A. Louveau, Descriptive Set Theory and Harmonic Analysis, The Journal of Symbolic Logic, 57(2):413441 (1992), surveying classical and modern connections with harmonic analysis; Toshiro Terano, Kiyoji Asai, and Michio Sugeno, Fuzzy Systems Theory and Its Applications (1992), a presentation of theory and of applications to various areas including image recognition and information retrieval; Antonio Di Nola, et al., Fuzzy Relation Equations and Their Applications to Knowledge Engineering (1989), an introduction; Harry Gonshor, An Introduction to the Theory of Surreal Numbers (1986), assuming some background in naive set theory; Wayne Blizard, The Development of Multiset Theory, Modern Logic, 1(4):319352 (1991), and Correction to The Development of Multiset Theory,' Modern Logic, 2(2):219 (1991), a broad survey with emphasis on recent developments; Stan Wagon, The Banach-Tarski Paradox (1985, reissued 1993), a presentation of several paradoxical theorems and related problems, with a discussion of the role played by AC; M. Bekkali, Topics in Set Theory (1991), developing several advanced topics; and Judith Roitman, The Uses of Set Theory, The Mathematical Intelligencer, 14(1):6369 (1992), demonstrating applications to areas of mathematics not generally considered to be closely related to set theory. Joseph Hashisaki Robert R. Stoll The Editors of the Encyclopdia Britannica