Naive Set Theory

Set Theory is the study of sets. The definition of a set in mathematics generally lines up with the typical meaning in everyday use - a collection of things. Examples include the first names of the band members of AC/DC, $\{\text{Angus}, \text{Malcolm}, \text{Cliff}, \text{Brian}, \text{Phil} \}$, the set of prime numbers less than ten, $\{2, 3, 5, 7\}$, and the Scandinavian countries, $\{\text{Norway}, \text{Sweden}, \text{Denmark}\}$. But before saying more about sets, it's important to note that set theory is generally split up between naive set theory and axiomatic set theory. The two approaches cover the same concepts but differ from the outset, starting with their definitions of sets.

Naive set theory is called "naive" because it does not have a philosophically rigorous definition of a set and instead presents it as an intuitive concept. The "collection of things" definition given above is as reasonable as any. This definition is nonetheless rigorous enough for a great many use cases. The benefit of naive set theory is that it gives most folks a powerful enough language of sets so that they can move on to studying other fields of mathematics that build on it. Perhaps surprisingly, nearly all other branches of what most folks would identify as math, from graph theory to complex analysis to differential topology, build on the concept of sets. The cost of naive set theory is that the intuitive definitions allow for the concoction of paradoxes, which is deeply unsettling to mathematicians whose one unifying trait across all disciplines is their metaphysical obsession with methodological rigor.

For example, consider the set of all sets that are not members of themselves and call it $S$. If $S$ is not a member of itself, it belongs to the set of all sets that do not contain themselves - but this is precisely $S$, which contradicts its own definition. This is known as Russell's Paradox, and is one of many paradoxes in naive set theory.

The good news is that most math gets on fine without ever needing to resolve these issues, but for those who insist on going all the way down to the bottom of the math pyramid, there's axiomatic set theory. Axiomatic set theory is not a single theory but refers to any number of rigorous definitions of sets. One of the most popular formulations is Zermelo-Frankel set theory, often abbreviated ZFC, where the C stands for the axiom of choice (but more on that axiom another time). Von Neumann-Bernays-Gödel (NBG) set theory is a related formulation, and still others exist as well.

Axiomatic set theory provides rigorous definitions of properties of and operations on sets (called axioms) that are formulated in first order logic, which itself covers the very structure of logical statements. The philosophical goal is to salvage all of the good things about naive set theory, such as unions, power sets, and induction, while getting rid of the bad things, such as paradoxes. Furthermore, expressing everything in first order logic allows computers to start working on mathematical proofs, although that it is also another topic for another time.

The choice of which form of set theory to study is usually one motivated by the economics of time. Many serious math students would love master everything like their boy von Neumann, but, absent Mephistopheles, this option is not on the table for the simple reason that study time is finite.

The best advice is to start with naive set theory and come back to axiomatic set theory if it either becomes necessary or it is in fact the subject that most interests you. The real numbers depend on the rational numbers, which in turn depend on the integers, which in turn depend on the natural numbers, and the natural numbers are a set of fundamental importance, but this only implies that sets themselves are of even more fundamental importance. How far down must a student go before they can say they've started doing "rigorous" math? It turns out that we have to dig our heels into the bedrock somewhere, below which more rigorous justifications aren't required, and naive set theory is the classic place to do it. Until then, you might ask, "but what is a real number, really?" and the answer is is a rigorous construction involving the rational numbers. You can in turn ask what a rational number is, and a similarly rigorous answer exists for that question as well. But by the time you ask what a set is, the rigorous answer is "Come on, man, stop being a wise guy."