Mathematical set theory is, to put it simply enough, the mathematical theory of sets.
It is the perfect definition, if one is not afraid of the classic "never use the word itself in the definition" taboo. This being said, there are better, more long-winded ways of defining set theory.
In essence, set theory revolves around the idea of grouping mathematical "sets" (is there a better word for them?) together by common features.
For instance, one set might include all integers, while another might include all negative numbers, while still others might include only the even, odd or prime numbers. As should be plainly obvious, there is really no limit to the number of sets possible to create in mathematics.
So what is the importance of any of this?
The Purpose of Set Theory
Set theory plays important roles in many aspects of mathematics, and as such has been given its very own set of symbols and terminology. Within the mathematics of sets (the most basic forms of which are usually taught in advanced algebra or pre-calculus courses), one may learn to define the following two important terms1:
Unions - A combination of two sets, including all members, either shared or separate. For instance, {1,2,3,4,5} is a union of {1,2,3} and {3,4,5}.
Intersections - A set consisting of all shared members of two or more other sets. Using the previous example, {3} is the entire intersection set of the sets {1,2,3} and {3,4,5}
These are elements of an aspect of set theory known as "membership." While it is easy to determine membership (also known as "set inclusion") in simple sets like these, there are certainly more complicated aspects of set theory which require much more intense mathematics, sometimes delving into some of the most fundamental aspects of number theory.
For instance, set theorists might ask a question regarding the intersection of {odd integers} and {prime integers}. This is especially difficult as both of these are infinite sets, meaning that there is no set number of elements to them - they can continue being listed forever. This idea of infinity is a major aspect of set theory, and has become the chief focus of many mathematicians over the years, leading to such astonishing, revolutionary and controversial notions as that of multiple "degrees" of infinity.
The Discovery of Set Theory
While the fundamental aspects of set theory seem relatively simple enough, and thus one might assume that these ideas have been around since the early days of mathematics itself, one of the astonishing facts about set theory is that one can point to a precise period of its beginning. A precise date, in fact. And a specific person.
That person was Georg Cantor. The Year was 1874.
That year, Cantor - a German mathematician - published a paper called "On a Characteristic Property of All Real Algebraic Numbers." While the concept of infinity had been toyed with for several millenia at this point, Cantor's paper marked the first realy, mathematical basis for beginning to acknowledge these difficult ideas in a fundamental and rigid way.
After Cantor, many other mathematicians grabbed hold of the reins of set theory, taking it in any number of new directions, not always mathematical.
Extension of Set Theory
Set theory can be used, in essence, to determine absolutely anything which can be divided up into groups (for instance, in statistical applications - finding the intersection of the sets {teenagers} and {people who watch X television show}, as an example)
Over the past century and a half, set theory has developed a rigid formalism that places it firmly as one of the more important mathematical fields of study today. It continues to grow as more and more people find new ways to look at number theory, the concept of infinity, and the very essence of how things can be counted.
References:
"Set Theory." Stanford Encyclopedia of Philosophy."
"The Colossal Book of Mathematics." Martin Gardner. W.W. Norton & Co. 2001.
 |
