Tourlakis G. Lectures in Logic and Set Theory. Vol 1. Mathematical Logic 2003

Textbook in PDF format

This two-volume work bridges the gap between introductory expositions of logic (or set theory) and the research literature. It can be used as a text in an advanced undergraduate or beginning graduate course in mathematics, computer science, or philosophy. The volumes are written in a user-friendly lecture style that makes them equally effective for self-study or class use. Volume I includes formal proof techniques, applications of compactness (including nonstandard analysis), computability and its relation to the completeness phenonmenon, and the first presentation of a complete proof of Godel's 2nd incompleteness since Hilbert and Bernay's Grundlagen.
Basic Logic
First Order Languages
A Digression into the Metatheory: Informal Induction and Recursion
Axioms and Rules of Inference
Basic Metatheorems
Semantics; Soundness, Completeness, Compactness
Substructures, Diagrams, and Applications
Defined Symbols
Computability and Uncomputability
Arithmetic, Definability, Undefinability and Incompletableness
Exercises
The Second Incompleteness Theorem
Peano Arithmetic
A Formal β-Function
Formal Primitive Recursion
The Boldface Arithmetization
Derivability Conditions; Fixed Points
Exercises