Halbeisen / Krapf Gödel's Theorems and Zermelo's Axioms

A Firm Foundation of Mathematics
Erscheinungsjahr 2020
ISBN: 978-3-030-52279-7
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 236 Seiten, eBook

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.

Zielgruppe

Graduate

Autoren/Hrsg.

Halbeisen, Lorenz

Krapf, Regula

Inhaltsverzeichnis

A Natural Approach to Natural Numbers.- Part I Introduction to First-Order Logic.- Syntax: The Grammar of Symbols.- Semantics: Making Sense of the Symbols.- Soundness & Completeness.- Part II Gödel’s Completeness Theorem.- Maximally Consistent Extensions.- Models of Countable Theories.- The Completeness Theorem.- Language Extensions by Definitions.- Part III Gödel’s Incompleteness Theorems.- Models of Peano Arithmetic and Consequences for Logic.- Arithmetic in Peano Arithmetic.- Gödelisation of Peano Arithmetic.- The Incompleteness Theorems.- The Incompleteness Theorems Revisited.- Completeness of Presburger Arithmetic.- Models of Arithmetic Revisited.- Part IV Zermelo’s Axioms.- Axioms of Set Theory.- Models of Set Theory.- Models of the Natural and the Real Numbers.- Tautologies.

Über Autor(innen)

Lorenz Halbeisen is Lecturer at the ETH Zürich since 2014.