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