Textbook in PDF format

There are many wonderful introductory texts on mathematical logic, but there are also many not-so-useful books. So how do you find your way around the very large literature old and new, and how do you choose what to read? Beginning Mathematical Logic provides the necessary guide. It introduces the core topics and recommends the best books for studying these topics enjoyably and effectively. This will be an invaluable resource both for those wanting to teach themselves new areas of logic and for those looking for supplementary reading before or during a university course.
Peter Smith was formerly Senior Lecturer in Philosophy at the University of Cambridge, and taught logic for more years than he cares to remember. His books include Explaining Chaos (1998), An Introduction to Formal Logic (2003; 2020), An Introduction to Gödel's Theorems (2007; 2013), and Gödel Without (Too Many) Tears (2020). He was also editor of Analysis for a dozen years.
Preface
The Guide, and how to use it
Who is the Guide for?
The Guide’s structure
Strategies for self-teaching from logic books
Choices, choices
So what do you need to bring to the party?
Two notational conventions
A very little informal set theory
Sets: a checklist of some basics
A note about naivety
Recommendations on informal basic set theory
Virtual classes, real sets
First-order logic
Propositional logic
FOL basics
A little more about types of proof-system
Basic recommendations for reading on FOL
Some parallel and slightly more advanced reading
A little history (and some philosophy too)
Postscript: Other treatments?
Second-order logic, quite briefly
A preliminary note on many-sorted logic
Second-order logic
Recommendations on many-sorted and second-order logic
Conceptual issues
Model theory
Elementary model theory
Recommendations for beginning first-order model theory
Some parallel and slightly more advanced reading
A little history
Arithmetic, computability, and incompleteness
Logic and computability
Computable functions
Formal arithmetic
Towards Gödelian incompleteness
Main recommendations on arithmetic, etc.
Some parallel/additional reading
A little history
Set theory, less naively
Set theory and number systems
Ordinals, cardinals, and more
Main recommendations on set theory
Some parallel/additional reading on standard ZFC
Further conceptual reflection on set theories
A little more history
Postscript: Other treatments?
Intuitionistic logic
A formal system
Why intuitionistic logic?
More proof theory, more semantics
Basic recommendations on intuitionistic logic
Some parallel/additional reading
A little more history, a little more philosophy
Elementary proof theory
Preamble: a very little about Hilbert’s Programme
Deductive systems, normal forms, and cuts
Proof theory and the consistency of arithmetic
Main recommendations on elementary proof theory
Some parallel/additional reading
Modal logics
Some basic modal logics
Provability logic
First readings on modal logic
Suggested readings on provability logic
Alternative and further readings on modal logics
Finally, a very little history
Other logics?
Relevant logic
Readings on relevant logic
Free logic
Readings on free logic
Plural logic
Readings on plural logic
Going further
A very little light algebra for logic?
More model theory
More on formal arithmetic and computability
More on mainstream set theory
Choice, and the choice of set theory
More proof theory
Higher-order logic, the lambda calculus, and type theory
Index of authors