Kaye R.W. The Mathematics of Logic: A Guide to Completeness Theorems and their Applications

Cambridge University Press, 2007. — 216 p. — ISBN: 052170877X, 9780521708777This undergraduate textbook covers the key material for a typical first course in logic, in particular presenting a full mathematical account of the most important result in logic, the Completeness Theorem for first-order logic. Looking at a series of interesting systems, increasing in complexity, then proving and discussing the Completeness Theorem for each, the author ensures that the number of new concepts to be absorbed at each stage is manageable, whilst providing lively mathematical applications throughout. Unfamiliar terminology is kept to a minimum, no background in formal set-theory is required, and the book contains proofs of all the required set theoretical results. The reader is taken on a journey starting with König's Lemma, and progressing via order relations, Zorn's Lemma, Boolean algebras, and propositional logic, to completeness and compactness of first-order logic. As applications of the work on first-order logic, two final chapters provide introductions to model theory and nonstandard analysis.How to read this book.
König's Lemma.
Two ways of looking at mathematics.
König’s Lemma and reverse mathematics.
Posets and maximal elements.
Introduction to order.
Zorn's Lemma and the Axiom of Choice.
Formal systems.
Post systems and computability.
Deductions in posets.
Proving statements about a poset.
Linearly ordering algebraic structures.
Boolean algebras.
Boolean algebra and the algebra of Boole.
Propositional logic.
A system for proof about propositions.
Decidability of propositional logic.
Valuations.
Semantics for propositional logic.
The complexity of satisfiability.
Filters and ideals.
Algebraic theory of boolean algebras.
Tychonov's Theorem.
The Stone Representation Theorem.
First-order logic.
Second- and higher-order logic.
Completeness and compactness.
Proof of completeness and compactness.
The Compactness Theorem and topology.
The Omitting Types Theorem.
Model theory.
Countable models and beyond.
Cardinal arithmetic.
Nonstandard analysis.
Infinitesimal numbers.
Overspill and applications.