Gödel's Theorems and Zermelo's Axioms

A Firm Foundation of Mathematics

Authors: Halbeisen, Lorenz, Krapf, Regula

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  • 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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  • ISBN 978-3-030-52279-7
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Hardcover $74.99
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About this Textbook

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.



About the authors

Lorenz Halbeisen is Lecturer at the ETH Zürich since 2014.


Table of contents (17 chapters)

Table of contents (17 chapters)

Buy this book

eBook $59.99
price for USA in USD
  • ISBN 978-3-030-52279-7
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • Immediate eBook download after purchase and usable on all devices
  • Bulk discounts available
Hardcover $74.99
price for USA in USD
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Bibliographic Information

Bibliographic Information
Book Title
Gödel's Theorems and Zermelo's Axioms
Book Subtitle
A Firm Foundation of Mathematics
Authors
Copyright
2020
Publisher
Birkhäuser Basel
Copyright Holder
Springer Nature Switzerland AG
eBook ISBN
978-3-030-52279-7
DOI
10.1007/978-3-030-52279-7
Hardcover ISBN
978-3-030-52278-0
Edition Number
1
Number of Pages
XII, 236
Topics