Burgess J. Set Theory. Elements in Philosophy and Logic 2022

Textbook in PDF format

Set theory is a branch of mathematics with a special subject matter, the infinite, but also a general framework for all modern mathematics, whose notions figure in every branch, pure and applied. This Element will offer a concise introduction, treating the origins of the subject, the basic notion of set, the axioms of set theory and immediate consequences, the set-theoretic reconstruction of mathematics, and the theory of the infinite, touching also on selected topics from higher set theory, controversial axioms and undecided questions, and philosophical issues raised by technical developments.
Historical Roots
The Notion of Set
The Zermelo–Fraenkel Axioms
Immediate Consequences
Number Systems Within Set Theory
Infinities
The Axiom of Choice
Topics in Higher Set Theory
Metamathematics of Set Theory
Large Cardinals and Determinacy
Concluding Philosophical Remarks