**Student Mathematical Library**

Volume: 89;
2019;
185 pp;
Softcover

MSC: Primary 03;

**Print ISBN: 978-1-4704-5272-8
Product Code: STML/89**

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $44.00

**Electronic ISBN: 978-1-4704-5407-4
Product Code: STML/89.E**

List Price: $55.00

AMS Member Price: $44.00

MAA Member Price: $44.00

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#### Supplemental Materials

# A First Journey through Logic

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*Martin Hils; François Loeser*

The aim of this book is to present mathematical logic to students who are interested in what this field is but have no intention of specializing in it. The point of view is to treat logic on an equal footing to any other topic in the mathematical curriculum. The book starts with a presentation of naive set theory, the theory of sets that mathematicians use on a daily basis. Each subsequent chapter presents one of the main areas of mathematical logic: first order logic and formal proofs, model theory, recursion theory, Gödel's incompleteness theorem, and, finally, the axiomatic set theory. Each chapter includes several interesting highlights—outside of logic when possible—either in the main text, or as exercises or appendices. Exercises are an essential component of the book, and a good number of them are designed to provide an opening to additional topics of interest.

#### Readership

Undergraduate and graduate students and researchers interested in learning basics of mathematical logic.

#### Reviews & Endorsements

This little book discusses, in six chapters totaling only about 170 pages of text, a number of facets of mathematical logic: set theory, the predicate calculus, model theory, recursive functions, Godel's Incompleteness Theorems, etc. The text will not make anybody an expert in these issues, but will at least provide a decent acquaintance with the basics at a mathematically correct level.

-- Mark Hunacek, MAA Reviews

#### Table of Contents

# Table of Contents

## A First Journey through Logic

- Cover Cover11
- Title page iii4
- Copyright iv5
- Contents v6
- Introduction ix10
- Chapter 1. Counting to Infinity 114
- Introduction 114
- 1.1. Naive Set Theory 114
- 1.2. The Cantor and Cantor-Bernstein Theorems 215
- 1.3. Orders 316
- 1.4. Operations on Orders 518
- 1.5. Ordinal Numbers 720
- 1.6. Ordinal Arithmetic 1124
- 1.7. The Axiom of Choice 1427
- 1.8. Cardinal Numbers 1427
- 1.9. Operations on Cardinals 1629
- 1.10. Cofinality 1932
- 1.11. Exercises 2235
- 1.12. Appendix: Hindman’s Theorem 2841

- Chapter 2. First-order Logic 3346
- Chapter 3. First Steps in Model Theory 6578
- Chapter 4. Recursive Functions 89102
- Introduction 89102
- 4.1. Primitive Recursive Functions 90103
- 4.2. The Ackermann Function 94107
- 4.3. Partial Recursive Functions 96109
- 4.4. Turing Computable Functions 98111
- 4.5. Universal Functions 107120
- 4.6. Recursively Enumerable Sets 109122
- 4.7. Elimination of Recursion 113126
- 4.8. Exercises 115128

- Chapter 5. Models of Arithmetic and Limitation Theorems 119132
- Introduction 119132
- 5.1. Coding Formulas and Proofs 120133
- 5.2. Decidable Theories 122135
- 5.3. Peano Arithmetic 125138
- 5.4. The Theorems of Tarski and Church 131144
- 5.5. Gödel’s First Incompleteness Theorem 133146
- 5.6. Definability of Satisfiability for Σ₁-formulas 134147
- 5.7. Gödel’s Second Incompleteness Theorem 137150
- 5.8. Exercises 141154

- Chapter 6. Axiomatic Set Theory 147160
- Introduction 147160
- 6.1. The Framework 147160
- 6.2. The Zermelo-Fraenkel Axioms 148161
- 6.3. The Axiom of Choice 155168
- 6.4. The von Neumann Hierarchy and the Axiom of Foundation 157170
- 6.5. Some Results on Incompleteness, Independence and Relative Consistency 161174
- 6.6. A Glimpse of Further Independence and Relative Consistency Results 169182
- 6.7. Exercises 173186

- Bibliography 179192
- Index 181194
- Back cover Back cover1201