I only started devoting myself to set theory in the second semester of my junior year, but thanks to various resources I was able to pick up the basics rather quickly. Here I list some of the books I’ve used (and am still using), and my brief view on their content.

Set Theory, Kenneth Kunen (2011)

This is the book I used when I first started learning set theory.

This book excelled at conveying highly abstract set-theoretic concepts in a very intuitive way. Set theory and topology are similar in many ways, and the book tries to remind the reader of such similarities from time to time. If you are a fan of analysis and have a background in topology (like me back in junior year), I believe this book is for you.

Unfortunately, this book uses a lot of nonstandard notations (e.g. $R(\alpha)$ instead of $V_\alpha$, $L(\alpha)$ instead of $L_\alpha$, $L(\alpha)[A]$ instead of $L_\alpha(A)$ etc.), so readers must be cautious of the trues standard notations when reading this book.

The exercises seemed too easy or difficult, and not many hints were provided.

I strongly recommend this book to people who want to pick up set theory.

Set Theory, Thomas Jech (2003)

This is the book that is always by my side nowadays.

This book is a set theory bible. Much of the theory in this book remains relevant nowadays.

Most of the book is very detailed and well-written but the first 6 chapters are too brief for beginners to study. I recommend beginners to start with Kunen (above), then this book from Chapter 7 onwards. However, Jech’s approach to forcing uses Boolean algebras. While the theory of forcing with Boolean algebras is extremely elegant, it is rarely used in practice, so I recommend readers refer to Kunen’s book (2011) for forcing instead. Regardless, it is worth learning forcing with Boolean algebras once.

This book has lots of exercises with good hints. They are also reasonably difficult and I recommend readers attempt them as much as possible.

The Higher Infinite, Akihiro Kanamori (2003)

This book is like Jech’s book but focuses on large cardinals.

I don’t read this book very often, but I feel that this book complements Jech’s book very well (i.e. in parts where Jech’s book is not clear, reading Kanamori’s book usually clarifies my doubts).

I only attempted a few exercises in this book.

Set Theory: An Introduction to Independence Proofs, Kenneth Kunen (1980)

This is the predecessor of Kunen’s 2011 Set Theory.

I don’t have much to say about this book, but compared to the 2011 version I feel that this book introduces concepts in a very unmotivated manner. I found it very difficult to get through Chapter 3.

Contrary to the opinions of many logicians I’ve talked to, I feel that the 2011 version is better than this one.

Combinatorial Set Theory: With a Gentle Introduction to Forcing, Lorenz J. Halbeisen (2018)

This is a very recent book on combinatorial set theory. It also serves as a bible, but (unsurprisingly) focuses on infinitary combinatorics. This book is especially helpful if you’re looking for various properties (e.g. proper, adds a dominating real) of common forcings. The proofs are detailed and helpful, and its recency also makes it one of the more updated bibles.

I strongly do not recommend starting on set theory, or just self-studying in general, with this book (yes I tried).