# Mathematical Logic

Mort Yao*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

Textbook:

- Herbert B. Enderton.
(*A Mathematical Introduction to Logic, 2nd edition.***AMIL**)- Author’s commentary (Internet Archive)
- Solutions
- Book notes by Peter Smith

Supplementary reading:

- 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.*

- Introduction
- Propositional logic (sentential logic)
- First-order logic
- Second-order logic
- Modal logic

# 2 Axiomatic set theory

Reading:

- 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 reading on the set theory: Serious set theory (also includes set theories other than ZFC, such as NBG)

# 3 Model theory

Reading:

- Maria Manzano.
*Model Theory.*

- Kripke semantics
- Algebraic logic

# 4 Computability theory

Reading:

- 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.*

- Number theory
- Presburger arithmetic
- Peano arithmetic

- Undecidability and Gödel’s incompleteness theorems

# 5 Structural proof theory

Reading:

- Sara Negri, Jan von Plato and Aarne Ranta.
*Structural Proof Theory.*

- Deep inference and cirquent calculi

# 6 Non-classical logic

Reading:

- 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.*

- Intuitionistic logic
- Intermediate logics
- Minimal logic

- Substructural logic and paraconsistent logic
- Linear logic
- Relevant logic

- Many-valued logic
- Fuzzy logic
- Probability logic

- Non-reflexive logic