A First Course in Mathematical Logic and Set Theory

CHAPTER 3

SET THEORY

3.1 SETS AND ELEMENTS

The development of logic that resulted in the work of Chapters 1 and 2 went through many stages and benefited from the work of various mathematicians and logicians through the centuries. Although modern logic can trace its roots to Descartes with his mathesis universalis and Gottfried Leibniz's De Arte Combinatoria (1666), the beginnings of modern symbolic logic is generally attributed to Augustus De Morgan [Formal Logic (1847)], George Boole [Mathematical Analysis of Logic (1847) and An Investigation of the Laws of Thought (1847)], and Frege [Begriffsschrift (1879), Die Grundlagen der Arithmetik (1884), and Grundgesetze der Arithmetik (1893)]. However, when it comes to set theory, it was Georg Cantor who, with his first paper, “Ueber eine Eigenschaft des Inbegriffs aller reellen algebraischen Zahlen” (1874), and over a decade of research, is the founder of the subject. For the next four chapters, Cantor's set theory will be our focus.

A set is a collection of objects known as elements. An element can be almost anything, such as numbers, functions, or lines. A set is a single object that can contain many elements. Think of it as a box with things inside. The box is the set, and the things are the elements. We use uppercase letters to label sets, and elements will usually be represented by lowercase letters. The symbol ∈ (fashioned after the Greek letter epsilon) is used to mean “element of,” so if A is a set and a is an element of A,