Set Theory

No one will expel us from this paradise Cantor has created for us.

Background

Most mathematical theories are nowadays presented using the language and notation of set theory. This is perhaps surprising since set theory only developed towards the close of the nineteenth century, and serious flaws were very soon discovered in it. Georg Cantor (1845-1918) is regarded as the founder of the discipline of set theory, though Richard Dedekind (1831-1916), influenced him greatly. Cantor developed this theory so as to be able to compare the sizes of various infinite sets of numbers. Cantor formulated set theory in an informal and intuitive way using seemingly obvious principles. This is still the set theory that is usually used, despite the flaws, which are politely called "paradoxes", but are actually contradictions - at least if the principles are taken as axioms. However, the use of the word "paradox" is defensible, because during the twentieth century several axiomatic formulations of set theory were created, which, so far as is known, are free from the contradictions.

Intuitively, a set is any collection of objects conceived of as a whole. The individual objects that make up the set are called the elements or members of the set (these terms are used interchangeably). Collecting objects together in this way and transferring our attention from the individuals to the whole collection is commonplace in everyday life: our language is full of such words as "family", "herd", "pack", "flock". In Cantor's definition the members of a set must also be "definite" and "separate" or "distinguishable". "Separate" is apparently intended to mean that when we want to count the elements of the set, we never count any particular element more than once - to be sure that we have not done so, we need an absolutely certain method of distinguishing between the elements.† This may be difficult. For example, we can conceive of the set of points more than six inches from a given point. But as distance cannot be measured exact­ly, the status of some points is uncertain. The term "definite" means that we must be unequivocally able to decide whether any particular object is or is not a member of the set.

We can form a set either by explicitly listing all its members, or, more usually, by specifying some property that the members share. In logic, a statement that asserts something about an object x is called a formula in x. For example the statement "x is a domestic cat" is a formula in x. Here x is a "place-holder" which we will replace with the actual name of some object. We shall probably expect "Tiddles is a domestic cat" to be a true assertion, but "Rover is a domestic cat" to be false. ‡A set might be defined using a formula which is never true. Such a set is called an empty set. For example the set defined by the formula "x is a living unicorn" is (pre­sumably) empty. The Intuitive Principle of Abstraction states that for any formula in x, we can form the set of all objects for which the formula is a true statement. The Principle of Abstraction seems plausible, but its unrestricted causes trouble: a set theory taking it as an axiom would yield contradictions*.*For example: Russell's paradox

Set Equality

Two sets are considered to be equal if (and only if) they have the same elements. For example the sets "The Smith Family" and the set "The inhabitants of 23 Acacia Avenue" might be equal even though they have been specified using a different property. Of course the properties I used to specify the last two sets are not mathematical at all, and would be highly ambiguous unless we have previously agreed on which Smith Family we and which 23 Acacia Avenue we are talking about. But in mathematics this concept of equality can be very useful, as we may be able to prove some property is true for a certain set by proving it for some apparently different set which we then prove has the same members as our first set. This definition of set equality is called the Intuitive Principle of Extension. With this definition of equality we see that there is in fact only one empty set.

Set Operations

You can find a more mathematical summary of the set theory needed for An Atlas of Sym­metry in Sets. Given two sets we can combine them in various ways. For example we can form their intersection - the set whose members are members of both the original sets, or the union, the set whose members are members of either of the original sets. Another very important way of combining two sets is to form the set of all pairs where the first element in the pair comes from the first set, and the second element from the second set. Although this may sound a little obscure it is a necessary step in building mathematical objects such as relations and functions.