This book is inspired by The Structure of Proof: With Logic and Set Theory published by Prentice Hall in 2002. My motivation for that text was to use symbolic logic as a means by which to learn how to write proofs. The purpose of this book is to present mathematical logic and set theory to prepare the reader for more advanced courses that deal with these subjects either directly or indirectly. It does this by starting with propositional logic and first-order logic with sections dedicated to the connection of logic to proof-writing. Building on this, set theory is developed using first-order formulas. Set operations, subsets, equality, and families of sets are covered followed by relations and functions. The axioms of set theory are introduced next, and then sets of numbers are constructed. Finite numbers, such as the natural numbers and the integers, are defined first. All of these numbers are actually sets constructed so that they resemble the numbers that are their namesakes. Then, the infinite ordinal and cardinal numbers appear. The last chapter of the book is an introduction to model theory, which includes applications to abstract algebra and the proofs of the completeness and compactness theorems. The text concludes with a note on Gödel's incompleteness theorems.
MICHAEL L. O'LEARY
Glen Ellyn, Illinois