Logic and Foundations

Below is a list of reviews of books on logic and foundations.

A mathematical introduction to logic
Enderton, H. B. (2nd ed. 2001). ISBN 9780122384523

The default introduction to mathematical logic. Good for beginning to intermediate undergrads, no knowledge of formal logic necessary, although prior exposure to elementary set theory (unions, intersections, etc) may be helpful. Relatively concise compared to Computability and Logic, and covers mostly only mathematical necessities.

Computability and Logic
Boolos, G. S., Burgess, J. P., Jeffery, R. C. (5th ed. 2007). ISBN 9780521701464.

Another good broad overview of logic. Approaches things from a theory of computational angle and covers more topics than Enderton. Directed at a broader audience, and readable to anyone who has some understanding of formal first order logic including philosophy and CS undergrads. Many chapters are skippable, and there is a lot of freedom in reading order. Topics range from elementary computability theory to Godel’s incompleteness and independence of several combinatorial results.

A Shorter Model Theory
Hodges, W. (1997). ISBN 9780521587136

An general introduction to model theory, the study of the semantics of formal logic. Assumes some abstract algebra, but some exposure to combinatorics or set theory will be helpful. This book has an intended audience of graduate students but is approachable as early as a 3rd year of undergraduate.

Reverse Mathematics
Stillwell, J. (2018). ISBN 9780691177175

A historical and mathematical introduction to the programs of arithmetization and reverse mathematics. Very short and concise text. Prior knowledge of real analysis is helpful. Topics covered range from the halting problem to analysis in various systems of arithmetic. The author has a clear vision of a narrative and follows it through, creating text that is a pleasure to read.

A Friendly Introduction to Mathematical Logic
Leary, C., Kristiansen, L., (2nd ed. 2015). ISBN 9781942341079

As the title says, this book is extremely friendly, being chatty to the point where some might find it slightly obnoxious, but I found it pretty refreshing. It covers a first course in mathematical logic up to the incompleteness theorem and its corollaries. It’s written in a very clear manner, and the only qualm I have about this book is that the propositional rule in the deduction system they use is somewhat confusing. Nevertheless, I found this to be a really good companion text for logic because whenever I was confused about an explanation in another book, the explanation in this book was invariably friendlier.

Modern Mathematical Logic
Mileti, J., (2022). ISBN 9781108833141

I personally find the notation very clean, and it is a perfect text for a first semester of mathematical logic up to the incompleteness theorems, with a smattering of other topics such as model theory and set theory thrown in. One thing that’s special about this book is that somewhat unusually, it starts with discussing the ideas in propositional logic, including the soundness and completeness theorems, which I found really helpful for understanding the corresponding ideas in first-order logic. Fun fact: It was originally named “A Mathematical Introduction to Mathematical Logic” before the author realized that every other logic book is also named that.