Tourlakis G. Lectures in Logic and Set Theory. Vol 2. Set Theory 2003

Textbook in PDF format

This volume contains the basics of Zermelo-Fraenkel axiomatic set theory. It is situated between two opposite poles: On one hand there are elementary texts that familiarize the reader with the vocabulary of set theory and build set-theoretic tools for use in courses in analysis, topology, or algebra – but do not get into metamathematical issues. On the other hand are those texts that explore issues of current research interest, developing and applying tools (constructibility, absoluteness, forcing, etc.) that are aimed to analyze the inability of the axioms to settle certain set-theoretic questions.
A Bit of Logic: A User’s Toolbox
The Set-Theoretic Universe, Naıvely
The Axioms of Set Theory
The Axiomof Choice
The Natural Numbers; Transitive Closure
Order
Cardinality
Forcing