Prerequisites: Most content on this topic assumes some mathematical proof techniques (incl. mathematical induction) and basic knowledge of naive set theory. The following books might be a good refresher:
- Daniel J. Velleman. How to Prove It: A Structured Approach.
- Paul Halmos. Naive Set Theory.
The first book also provides a fair introduction to propositional (sentential) logic.
1 Classical Logic
- Herbert B. Enderton. A Mathematical Introduction to Logic, 2nd edition. (AMIL)
- Elliott Mendelson. Introduction to Mathematical Logic, 4th edition.
- Stephen Cole Kleene. Introduction to Metamathematics.
- Stephen Cole Kleene. Mathematical Logic.
- Raymond M. Smullyan. First-Order Logic.
- Robert S. Wolf. A Tour Through Mathematical Logic.
1.1 Propositional Logic
1.2 First-Order Logic
1.3 * Second-Order Logic
1.4 * Modal Logic
2 Axiomatic Set Theory (ZFC)
- Herbert B. Enderton. The Elements of Set Theory.
- Raymond M. Smullyan and Melvin Fitting. Set Theory and the Continuum Problem.
- Thomas Jech. Set Theory, 3rd millennium edition.
See also Peter Smith’s suggestions for readings on the set theory: Serious set theory (also includes alternative set theories like NBG)
3 Model Theory
- Maria Manzano. Model Theory.
4 Computability Theory
- Peter Smith. An Introduction to Gödel’s Theorems.
- Richard Epstein and Walter Carnielli. Computability: Computable Functions, Logic, and the Foundations of Mathematics.
- George Boolos, John P. Burgess and Richard Jeffrey. Computability and Logic.
4.1 Peano Axioms
4.2 Gödel’s Incompleteness Theorems
5 Structural Proof Theory
6 Non-Classical Logic
- Graham Priest. An Introduction to Non-Classical Logic: From If to Is, 2nd edition.
- John L. Bell, David DeVidi and Graham Solomon. Logical Options: An Introduction to Classical and Alternative Logics.