# Gödel's Theorems and Zermelo's Axioms

## A Firm Foundation of Mathematics

• Lorenz Halbeisen
• Regula Krapf
Textbook

1. Front Matter
Pages i-xii
2. ### Introduction to First-Order Logic

1. Front Matter
Pages 5-5
2. Lorenz Halbeisen, Regula Krapf
Pages 7-18
3. Lorenz Halbeisen, Regula Krapf
Pages 19-34
4. Lorenz Halbeisen, Regula Krapf
Pages 35-44
3. ### Gödel’s Completeness Theorem

1. Front Matter
Pages 45-45
2. Lorenz Halbeisen, Regula Krapf
Pages 47-52
3. Lorenz Halbeisen, Regula Krapf
Pages 53-64
4. Lorenz Halbeisen, Regula Krapf
Pages 65-70
4. ### Gödel’s Incompleteness Theorems

1. Front Matter
Pages 71-71
2. Lorenz Halbeisen, Regula Krapf
Pages 73-78
3. Lorenz Halbeisen, Regula Krapf
Pages 79-88
4. Lorenz Halbeisen, Regula Krapf
Pages 89-108
5. Lorenz Halbeisen, Regula Krapf
Pages 109-121
6. Lorenz Halbeisen, Regula Krapf
Pages 123-136
7. Lorenz Halbeisen, Regula Krapf
Pages 137-149
5. ### The Axiom System ZFC

1. Front Matter
Pages 151-151
2. Lorenz Halbeisen, Regula Krapf
Pages 153-171
3. Lorenz Halbeisen, Regula Krapf
Pages 173-187
4. Lorenz Halbeisen, Regula Krapf
Pages 189-197
5. Lorenz Halbeisen, Regula Krapf
Pages 199-202
6. Lorenz Halbeisen, Regula Krapf
Pages 203-221
6. Back Matter
Pages 223-236

### Introduction

This book provides a concise and self-contained introduction to the foundations of mathematics. The first part covers the fundamental notions of mathematical logic, including logical axioms, formal proofs and the basics of model theory. Building on this, in the second and third part of the book the authors present detailed proofs of Gödel’s classical completeness and incompleteness theorems. In particular, the book includes a full proof of Gödel’s second incompleteness theorem which states that it is impossible to prove the consistency of arithmetic within its axioms. The final part is dedicated to an introduction into modern axiomatic set theory based on the Zermelo’s axioms, containing a presentation of Gödel’s constructible universe of sets. A recurring theme in the whole book consists of standard and non-standard models of several theories, such as Peano arithmetic, Presburger arithmetic and the real numbers.

The book addresses undergraduate mathematics students and is suitable for a one or two semester introductory course into logic and set theory. Each chapter concludes with a list of exercises.

### Keywords

mathematical logic set theory completeness theorem incompleteness theorem non-standard models Peano arithmetic Presburger arithmetic constructible universe

#### Authors and affiliations

• Lorenz Halbeisen
• 1
• Regula Krapf
• 2
1. 1.Departement MathematikETH ZürichZürichSwitzerland
2. 2.Institut für MathematikUniversität Koblenz-LandauKoblenzGermany