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Foundations and Computation

Introductory

Online resources at Logic Matters by Peter Smith

Kleene, Introduction to Metamathematics

Tennant, Natural logic

Chiswell and Hodges, Mathematical logic

Leary and Kristiansen, A friendly introduction to mathematical logic

Westerstahl, Foundations of logic

Rautenberg, A concise introduction to mathematical logic

Intermediate

Kunen, The Foundations of mathematics

Curry, Foundations of mathematical logic

Monk, Mathematical logic

Mendelsohn, Introduction to mathematical logic

Schoenfield, Mathematical logic

Robbin, Mathematical logic: a first course

Enderton, A mathematical introduction to logic

Ebbinghaus et al, Mathematical logic

van Dalen, Logic and structure

Johnstone, Notes on logic and set theory

Hodel, An introduction to mathematical logic

Manin, A course in mathematical logic for mathematicians

Advanced

Model theory

Chang and Keisler, Model theory

Hodges, Model theory

Marker, Model theory: an introduction

Tent and Ziegler, A course in model theory

Libkin, Elements of finite model theory

Kaye, Models of Peano arithmetic

Smullyan, Godel's incompleteness theorems

Proof theory

Takeuti, Proof theory

Troestra and Schwichtenberg, Basic proof theory

Girard, Proof theory and logical complexity, vols. 1 &2

Schwichtenberg and Wainer, Proofs and computations

Set theory

Levy, Basic set theory

Drake, Set theory: an introduction to large cardinals

Fraenkel et al, Foundations of set theory

Kunen, Set theory

Jech, Set theory

Smullyan and Fitting, Set theory and the continuum problem

Bell, Set theory: boolean-valued models and independence proofs

Kanamori, The higher infinite: large cardinals in set theory from their beginnings

Weaver, Forcing for mathematicians

Herrlich, Axiom of choice

Jech, The axiom of choice

Bell, The axiom of choice

Forster, Set theory with a universal set

Computation and complexity

Cutland, Computability: An introduction to recursive function theory

Cooper, Computability theory

Sipser, Introduction to the theory of computation

Goldreich, P, NP and NP-completenss

Arora and Barak, Computational complexity: a modern approach

Pudlak, Logical foundations of mathematics and computational complexity

The Curry-Howard correspondence and Lambda calculus

Hankin, An introduction to the Lambda calculus for computer scientists

Barendregt, The Lambda calculus: its syntax and semantics

Sorensen and Urzyczyn, Lectures on the Curry-Howard isomorphism

Hindley and Seldin, Lambda calculus and combinators: an introduction

Type theory

Nederpelt and Geuvers, Type theory and formal proof: an introduction

Bell, Higher order logic and type theory