Overview
- Provides a detailed proof of both of Gödel’s Incompleteness Theorems without building on recursion theory
- Presents detailed constructions of several standard and non-standard models of Peano Arithmetic, Presburger Arithmetic, Zermelo Fraenkel set theory and the real numbers
- Contains a self-contained and concise introduction into mathematical logic and axiomatic set theory which requires almost no prerequisites, whose only assumption is the notion of finiteness
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Table of contents (17 chapters)
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Front Matter
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Introduction to First-Order Logic
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Front Matter
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Gödel’s Completeness Theorem
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Front Matter
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Gödel’s Incompleteness Theorems
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Front Matter
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The Axiom System ZFC
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Front Matter
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Keywords
About this book
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.
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Authors and Affiliations
About the authors
Bibliographic Information
Book Title: Gödel's Theorems and Zermelo's Axioms
Book Subtitle: A Firm Foundation of Mathematics
Authors: Lorenz Halbeisen, Regula Krapf
DOI: https://doi.org/10.1007/978-3-030-52279-7
Publisher: Birkhäuser Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2020
Hardcover ISBN: 978-3-030-52278-0Published: 17 October 2020
Softcover ISBN: 978-3-030-52281-0Published: 17 October 2021
eBook ISBN: 978-3-030-52279-7Published: 16 October 2020
Edition Number: 1
Number of Pages: XII, 236