Chapter 5 Elements of set theory
Mathematicians treat the concepts of set and element as undefined primitives. Rules (in the form of axioms and axiom schemata) are provided for the manipulation of these objects. These extend the rules of first-order predicate calculus. In most foundational schemes (including the one presented below), absolutely every mathematical object — every number, every polynomial, every element of every set — is a set. So when we write \(x\in X\text{,}\) both \(x\) and \(X\) are sets.
The student who wants to go very deep into the subject of set theory should consult the astonishing text of Jech\autocite{Jech:2003tt}. The student who would prefer to work up to Jech's text should begin with Halmos's text\autocite{MR0453532}.
This course won't require any set theory beyond Halmos's book, but because general topology and set theory interact in various nontrivial ways, it would be intellectually dishonest not to give at least a quick overview of some the basic elements of set theory.