A Mathematical Introduction to Logic

Rating:
Summary: Terrific Book
Review: Enderton's writing is the best I've seen in any introductory math textbook; he is lucid, well organised, comfortably paced but free of expository flab. The exercises (judging from chapters 2 and 3) are not terribly difficult, but quite useful in building one's intuition and connecting logic to other mathematics. I had the book for my Logic class as a first-semester sophomore with very little experience with proofs and no abstract algebra, and found it quite accessible. I guess the book starts off with an advantage, being about a subject as interesting as logic, but that does not seriously detract from its merit.

Rating:
Summary: Very Good.
Review: I read the FIRST EDITION. This is definitely the best introductory mathematical logic book I've seen. It's the most rigorous, most advanced (a reasonably strong form of Godel's theorem is given), and is well-organized and very clearly written. It would be suitable as 1)an introduction for students with some mathematical experience- say a little abstract algebra and perhaps some previous exposure to logic. 2)a refresher for advanced students 3)a nice reference for basic topics. The exposition is great- Enderton always clearly explains what he's doing and why, keeping the reader focused on the big picture while going through the details. He helps to place topics in perspective, and has organized the book so readers can skip some of the more involved proofs and sections on the first reading. Chapter 1 covers propositional logic, with a general-purpose discussion of inductively defined sets, unique readability, and recursion. Many books these days do a sloppy job justifying recursive definition, or dont bother at all- Enderton does it right and is fairly detailed. Chapter 2 begins first order logic and has the most detailed proof of the completeness theorem I've seen. Sect 2.7 concerns translating between theories in different languages, something i hadnt seen developed explicitly before. 2.8 is a great exposure to nonstandard analysis- long enough to give you an idea how it works and why its useful. Chapter 3 begins with an analysis of some reducts of number theory- (N,0,S) (N,0,S,<) and (N,0,S,<,+) and shows how to eliminate quantifiers in them. Next, toward Godel's theorem, a finite set of axioms for a subtheory of number theory is given, and a host of relations and functions are shown to be representable in this theory. In 3.5 we get the fixed-point theorem, Tarskis thm, a weak Godels thm, a stronger Godels thm, and Church's Undecidability thm, and an introduction to the arithmetic hierarchy. 3.6 lifts Godels thm to show set theory is incomplete, and discusses Godels 2cd thm. Chap 4 is 2cd order logic, skolem normal form, many-sorted logic (a first order logic with different sets of variables ranging over different universes), and general 2cd order logic (restrictions are placed on the subsets "X" ranges over in the 2cd order formula \all X \phi). Basic recursion theory is developed throughout the book- Enderton begins with informal notions of computation, then defines a relation R as recursive iff it is representable in some consistent finitely axiomatizable theory, and discusses Church's thesis. 3.8 quickly covers universal computers, partial functions, Kleene normal form, unsolvability of the halting problem, the smn thm, Rice's them, and a register machine model. All this seemed a bit disorganized, so familiarity with computation and automata theory would be a plus. Heres the contents for the first edition, c1972:

Chapter Zero - USEFUL FACTS ABOUT SETS . . . .1
Chapter One - SENTENTIAL LOGIC
1.0 Informal Remarks on Formal Languages 14
1.1 The Language of Sentential Logic . . . . . 17
1.2 Induction and Recursion . . . . . . . . .22
1.3 Truth Assignments . . . . . . . . . . . .30
1.4 Unique Readability . . . . . . . . . . .39
1.5 Sentential Connectives . . . . . . . . . .44
1.6 Switching Circuits . . . . . . . . . . . .53
1.7 Compactness and Effectiveness . . . . . 58
Chapter Two - FIRST-ORDER LOGIC
2.0 Preliminary Remarks . . . . . . . . . .65

2.1 First-Order Languages . . . . . . . . . .67
2.2 Truth and Models . . . . . . . . . . . 79
2.3 Unique Readability . . . . . . . . . . . 97
2.4 A Deductive Calculus . . . . . . . . . .101
2.5 Soundness and Completeness Theorems . .124
2.6 Models of Theories . . . . . . . . . . . 140
2.7 Interpretations between Theories . . . ... 154
2.8 Nonstandard Analysis . . . . . . . . . . .164
Chapter Three - UNDECIDABILITY
3.0 Number Theory . . . . . . . . . . . . 174
3.1 Natural Numbers with Successor . . . . 178
3.2 Other Reducts of Number Theory . . . . 184
3.3 A Subtheory of Number Theory . . . . . . 193
3.4 Arithmetization of Syntax . . . . . . . . .217
3.5 Incompleteness and Undecidability . . . 227
3.6 Applications to Set Theory . . . . . . . .239
3.7 Representing Exponentiation . . . . . . .245
3.8 Recursive Functions . . . . . . . . . . .251
Chapter Four - SECOND-ORDER LOGIC
4.1 Second-Order Languages . . . . . . . . . 268
4.2 Skolem Functions . . . . . . . . . . . . 274
4.3 Many-Sorted Logic . . . . . . . . . . . . 277
4.4 General Structures . . . . . . . . . . . . 281
Index . . . . . . . . . . . . 291

Rating:
Summary: Excellent Textbook with lots of examples
Review: I used this book for self study of Mathematical Logic with the aim of understanding Godel's incompleteness theorem. I also referred to other introductory Mathematical Logic books. In my opinion, this book is by far the best among them. Very readable and contains lots of carefully selected examples.

Rating:
Summary: Excellent Textbook with lots of examples
Review: I used this book for self study of Mathematical Logic with the aim of understanding Godel's incompleteness theorem. I also referred to other introductory Mathematical Logic books. In my opinion, this book is by far the best among them. Very readable and contains lots of carefully selected examples.

Rating:
Summary: Excellent introduction to logic
Review: One of the very best introductions to logic, combining readability and depth. An excellent book.

Rating:
Summary: Great Book
Review: This is a great introductory book. Some set theory, sentential logic, first-order logic, metatheory/model theory,number theory, undecidability and Godel's Incompleteness, and Second-Order Logic. You still have to take a lot of time trying to soak in the stuff, but that's because of the complex nature of the material, not the book. The book itself is really good.