Where To Start In Pure Mathematics? Set Theory

So you'd like to learn pure mathematics. Where should you start? Real analysis? Topology? Number Theory? You can find introductory, undergraduate, preliminary, and even "gentle" textbooks written on any topic can think of, and it's not obvious that one subject "comes before" another the same way that algebra comes before pre-calculus. Likewise, plenty of syllabi from august universities claim that their courses are "students' first introduction to rigorous mathematics," only for others to puzzlingly claim "no prerequisites." What's the deal, then? Which subject you really study first?

The answer is set theory. Oddly enough, it's likely that set theory is at most a single course at most universities, rarely intended as an introductory pure mathematics course, and typically hidden in the course catalog by the profusion of analysis, topology, and abstract algebra courses. So then why is set theory the appropriate starting point? Because it's secretly the first subject taught in each of those classes. In fact, every standard text in higher mathematics starts with or assumes knowledge of set theory.

Don't believe me? Let's take a look at the canon of math textbooks for "real" math:

  • Real Analysis: Principles of Mathematical Analysis, Rudin: The first definition in Chapter 1 reads "If $A$ is a set (whose elements may be numbers or any other objects), we write $x \in A$ to indicate that $x$ is a member (or an element) of $A$." The next few pages are dedicated to the basics of sets. In fact, Chapter 2, "Basic Topology" (of $\mathbb{R}$) starts off with a section on "Finite, Infinite, and Uncountable Sets," and a number of the later topological proofs in the chapter in fact bring in set theory directly, and some in fact have nothing to do with metric topology.
  • Topology: Topology, James Munkres: Chapter 1 is entitled "Set Theory and Logic" and is over 70 pages long.
  • Abstract Algebra: Abstract Algebra, Dummit and Foote: Set theory appears before the first chapter in "Preliminaries." The first sentence in the affectionately numbered section 0.1 reads "The basics of set theory: sets, $\cap$, $\cup$, $\in$, etc. should be familiar to the reader." Nonetheless, the authors spend three and a half pages rehashing some basic definitions.
  • Graph Theory: Graph Theory, Bondy and Murty: After a perfunctory introductory paragraph, Chapter 1.1 "Graphs and Their Representations" trots out its first definition in terms of sets: "A graph $G$ is an ordered pair $(V(G), E(G))$ consisting of a set $V(G)$ of vertices and a set $E(G)$, disjoint from $V(G)$, of edges..."
  • Linear Algebra: Linear Algebra Done Right, Axler: The first sentence of Chapter 1 section A is "You should already be familiar with basic properties of the set R of real numbers." Immediately following that paragraph is the set-theoretic definition of the complex numbers in terms of the real numbers, complete with the curly braces and colon notation. Section 1.B provides a definition of vector spaces that begins "A vector space is a set $V$...".  (Oddly enough, this book is easiest one in the list, yet it assumes more knowledge of set theory than the others!)

You get the idea. Even if set theory doesn't get pride of place in the course catalog, all the textbooks seem to agree that knowledge of sets is critical to the "actual" topic at hand.

So what the heck is set theory? Set theory is the study of (wait for it) sets. A set is just a collection of objects. The letters of the alphabet form a set, the set of chairs at your kitchen table, the set of books on the top shelf of your bookcase, and so on. Due to their simplicity, sets form the basis for the other "higher" branches of math. Algebraic objects such as groups, rings, and fields are sets on which functions with particular properties are defined. "Spaces" such as vector spaces, topological spaces, measure spaces, and probability spaces are defined in terms of several sets that also meet additional requirements. And graphs, as we saw above, are defined in terms of a set of vertices and a set of edges between them.

Set theory also covers many of the basic building blocks of math, such as relations, functions, orders, and cardinality (aka "size"). And no course in set theory would be complete without a construction of the natural numbers from sets. Studying set theory therefore gives you a firm grasp on all of the loose tidbits that are called upon at will in other higher order subjects.

Constructions aside, set theory also has very little to do with numbers, which makes it perfect for learning proofs. In contrast, subjects like real analysis are a terrible fit for the job, because all of the theorems it wants to prove have been taken as The Obvious Truth by students for years at that point, needlessly complicating the immensely important task of learnings how to do proofs themselves. Learning how to write a proof is a herculean undertaking wholly independent of any actual subject, and it's the big kids equivalent of learning algebra. If you don't "get" algebra, your math career dies around ninth grade, and if you don't "get" proofs, your math career never leaves the collegiate frisbee field.

One final benefit of set theory is that it does not induce the latent fear of yet more fundamental math. If you start with real analysis, you'll look at the definitions and ask "Well shouldn't I study fields first? Or perhaps rational numbers?" Both questions lead to a death spiral of looking for The Beginning. Well, that's set theory. If you want to study math from the bottom up, set theory is square one. Philosophers will argue that logic is even more fundamental, but we can unironically place basics of first order logic as the chapter 0 to our own book on set theory.

So there you have it. Set theory is your pure math boot camp. Get proving!