### Topic Page: Set Theory

**SET THEORY**from

*Elsevier's Dictionary of Psychological Theories*

The Russian-born German mathematician set theory as a result of his examination of the circumstances in which a mathematical function is represented by a unique Fourier series (i.e., a generalization that any complex periodic pattern may be viewed as a particular sum of a number of sine waves). Whereas previous investigators had provided results for functions that are continuous on a given interval, Cantor considered set of points at which functions behave in a way that makes their Fourier series inappropriate. Cantor found that he could repeat this construction and obtain from one such set another set, sometimes indefinitely (cf., extension theorem of semantic entailment - in logic, a theorem in propositional calculus stating that if a set of premises p entails a conclusion q, then the addition of further premises from a larger set s that includes p cannot affect the truth of the conclusion q; this theorem is at the basis of the notion of monotonicity in logic, stating that a valid argument cannot be made invalid, nor an invalid argument made valid, by adding new premises). Cantor's approach led to a highly original arithmetic of the infinite, extending the concept of cardinal and ordinal numbers to infinite sets. Basic to Cantor's set theory is the idea that infinite sets have the same size, or cardinality, if and only if there is a one-to-one relationship between their members. Cantor demonstrated that the set of real numbers is “uncountable” (i.e., it cannot be formed in a one-to-one relationship with the set of integers), and that the set of subsets of a set is always larger than the original set. Cantor proposed - but could not solve - the problem of characterizing the cardinality of the continuum; such a problem is considered to be unsolvable in a more precise form. Other features of Cantor's theory of sets have become essential in the areas of topology and modern analysis in mathematics and statistics in psychology. Around 1900, Cantor and his friend Julius (1831-1916) simultaneously developed a naïve theory of sets to serve as a foundation for mathematics. Sets, or collections of objects, are represented typically by an upper-case letter or by a pair of brackets enclosing all of its members; for example, the set of “natural numbers” is: N = [1,2,3 ….], and the set of “black American presidents” is: F =Ø(this latter set is called a “null set” or “empty set”). Set theory inherently contains an interesting logical inconsistency called Russell's paradox [named after its enunciation in 1901 by the Welsh philosopher (1872-1970)] that centers on the idea that some sets are members of themselves and others are not. The so-called barber's paradox points out the paradox or inconsistency via an example: suppose there is a town barber who shaves all and only those men who do not shave themselves - from this it follows logically that if this barber shaves himself,then he does not, and if he does not, then he does (!) See also BOOLEAN SET THEORY; FOURIER'S LAW/SERIES/ANALYSIS; FUZZY SET THEORY;MIND/MENTAL SET, LAW OF; NEURAL NETWORK MODELS OF INFORMATION PROCESSING.

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