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), along with my brief views on their content.

Note. These were my views on the respective books as an undergraduate. I have briefly included my revised opinions as a fourth-year PhD student.

Set Theory, Kenneth Kunen (2011)

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

This book excels at conveying highly abstract set-theoretic concepts in an intuitive manner. Set theory and topology share many similarities, and the book occasionally reminds the reader of these connections. If you are a fan of analysis and have a background in topology (like I did back in my junior year), I believe this book is for you.

However, this book uses many 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 careful about the standard notations when reading it.

The exercises tend to be either too easy or too difficult, and not many hints are provided.

I strongly recommend this book to anyone interested in learning set theory.

Opinion as a PhD student. Nowadays, I rarely refer to this book, but I still believe that this is an excellent book to begin one’s journey in set theory.

Set Theory, Thomas Jech (2003)

This is the book that is always by my side these days.

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 six chapters are too brief for beginners to study. I recommend that beginners start with Kunen (above) and then proceed with 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; therefore, I recommend that readers refer to Kunen’s book (2011) for a more comprehensive treatment of forcing. Regardless, it is worth learning forcing with Boolean algebras at least once.

This book contains numerous exercises with helpful hints. They are also reasonably difficult, and I recommend that readers attempt as many of them as possible.

Opinion as a PhD student. I still refer to this book regularly.

The Higher Infinite, Akihiro Kanamori (2003)

This book is similar to Jech’s book, but it focuses on large cardinals.

I don’t read this book very often, but I feel that it 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 attempted only a few exercises from this book.

Opinion as a PhD student. I rarely refer to this book these days, as I rarely touch on large cardinals in my research. However, I like its treatment of determinacy Chapter 6.

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 it introduces concepts in a somewhat unmotivated manner. I found it very difficult to get through Chapter 3.

Contrary to the opinions of many logicians I’ve spoken to, I believe that the 2011 version is superior to this one.

Opinion as a PhD student. I don’t think I’ve opened this book since my undergraduate days.

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 the book’s recency also makes it one of the more updated bibles.

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

Opinion as a PhD student. This is the book that I refer to the most these days. It is an extremely helpful guide that details various important basic properties of different types of forcings. While I still do not think this book is suitable for beginners, I strongly recommend PhD students in set theory to use this book as a regular reference.