One of the primary features of mathematics is its transformation of the workings of nature into symbolic form. After systematically examining, manipulating and analyzing these abstract symbols, hopefully one may obtain a deeper understanding of the world. Perhaps the best example of this approach is the use of set theory. Created by Georg Cantor in the 1870s as an outgrowth of his study of the concept of infinity and transfinite numbers, set theory can be applied to a huge range of phenomena.
Sets are collections of things or ideas. A box of chocolates, a flock of geese, a pack of cards, and the collection of numbers 1 to 10 are all examples of sets. The things that make up a set are the members or elements of that set. Some things may be members of one set but not members of another. Describing the way set elements relate to each other, or to elements of another set, are the properties of that set. For instance, a box of chocolates is a member of the set of all boxes of chocolates, which is itself a member of the set of all things that are boxed. The box of chocolates is not, however, a member of the set of all guitar players. Some sets have a finite number of elements, such as the set of all people who have walked on the moon. The set of integers, on the other hand, is an infinite set.
Determining how, or where, different sets overlap or share the same elements is an important problem-solving technique in set theory. One of the more convenient uses for this procedure is in logic. For instance, the syllogism "All high-school students are younger than fifty years old; Jane is a high-school student; therefore Jane is younger than fifty years old" can be proved valid by properly constructing set diagrams, such as those invented by Leonhard Euler in the 1700s and John Venn (1834-1923) in the nineteenth century. More subtle is the statement "All high-school students are younger than fifty years old. Jane is under fifty years old. Therefore, Jane is a high-school student." This syllogism is invalid since not all people under fifty years old are necessarily high-school students. False formulations of this type are easy enough to spot without invoking set theory, but much more complex statements in semantics and number theory would be impossible to resolve without recourse to a rigorous procedure. This type of situation arises frequently in computer programming in which the information flow through a complex algorithm creates errors and logic tangles that are beyond the human mind fully to grasp and manipulate without an appropriate mathematical tool.
One of the most generally-known contributors to set theory was Charles Lutwidge Dodgson, known to the general public as Lewis Carroll, the author of Alice in Wonderland and Through the Looking Glass. Dodgson introduced the concept of the universal set, that is to say the set of all sets. When it is not possible to calculate the probability that a given item is an element of a set, it may be possible to calculate the probability that it is not an element of that set by means of Dodgson's concept of the universal set.
Several mathematicians used the methods of set theory to probe into the logic-structures that many felt were at the foundation of all mathematics, In 1904 Ernst Zermelo (1871-1953) used set theory to devise axioms that could be used to test the completeness of formal systems. Cantor's work with the theory of infinite sets and further work by Bertrand Russell on the logical foundations of mathematics led to the formulation by Russell and Alfred Whitehead of their epoch-making work Principia Mathematica. This work brought to a culmination the work of scores of mathematicians and thousands of years of mathematical effort by attempting to create a grand logical theory of arithmetic reminiscent, in scale, of Euclid's great work, the Elements. But not all mathematicians believed that this approach would bear fruit. Luitzen Egbertus Jan Brouwer, a Dutch mathematician, felt that it was a waste of time to speculate on the vague properties of such abstractions as the infinite set of natural numbers. According to Brouwer, it is meaningful to analyze only those mathematical structures that are introduced by a coherent method of construction. In 1920 Kurt Gödel resolved this debate by stunning the mathematical world with his proof of the incompleteness of formal systems, sweeping aside the approach taken by Russell, Whitehead and David Hilbert.
Despite these difficulties and the paradoxes that have arisen from set theory, the theory has found important applications in most branches of mathematics, statistics, and science. Perhaps least well-known are mathematical applications used in the works of modern composers who employ set theory to create sets of notes which serve the same role as melody and which are permutated according to mathematical laws into large-scale musical structures.
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