An Introduction to Naive Set Theory and The Concept of Innity: Guided by an Essay of Richard Dedekind

During the last 100 to 150 years, it has become common practice for mathematicians to consider collecting numbers (and/or other objects) into a single conglomerate. This point of view in fact permeates much of higher level mathematics. As such, students of mathematics should have a …rm grip on the underlying ideas of how these conglomerates behave. These conglomerates are typically called sets in modern mathematical language. This project introduces the intuitive or naive point of view on sets. In fact our exploration will be guided by the famous mathematician Richard Dedekind (1831-1916). Before continuing, it is important to point out that a rigorous de…nition of a set, or as Dedekind says a system of elements, is usually avoided by textbooks. This is for a couple of reasons. First, many early attempts at de…ning what a set is or should be lead to antinomies or paradoxes; Dedekind’s de…nition is not an exception. Despite this, Dedekind’s description of the behavior of sets is eloquent, enlightening, and it exposes essential mechanics of sets that a student should learn for further work in mathematics. Second, the study of set theory is a special topic in …rst order predicate logic and/or model theory, both of which are not typically in the early undergraduate curriculum. More speci…cally the topics addressed herein include sets, subsets, union and intersection of sets, functions, composition, injectivity, forward image and inverse of a function, set equivalence, and last but certainly not least, the ideas of in…nite and …nite are introduced. Furthermore, in this project properties and interconnections between these topics are also explored.