Gödel's Theorems and Zermelo's Axioms

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Sprache: Englisch

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Beschreibung

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 Zermelös 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.

Über den Autor

Lorenz Halbeisen is Lecturer at the ETH Zürich since 2014.

Zusammenfassung

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

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.

Details

Erscheinungsjahr:	2021
Fachbereich:	Grundlagen
Genre:	Mathematik , Medizin , Naturwissenschaften , Technik
Rubrik:	Naturwissenschaften & Technik
Medium:	Taschenbuch
Inhalt:	xii 236 S.
ISBN-13:	9783030522810
ISBN-10:	3030522814
Sprache:	Englisch
Einband:	Kartoniert / Broschiert
Autor:	Krapf, Regula Halbeisen, Lorenz
Auflage:	1st edition 2020
Hersteller:	Springer Nature Switzerland Springer International Publishing Springer International Publishing AG
Verantwortliche Person für die EU:	Springer Basel AG in Springer Science + Business Media, Heidelberger Platz 3, D-14197 Berlin, juergen.hartmann@springer.com
Maße:	235 x 155 x 14 mm
Von/Mit:	Regula Krapf (u. a.)
Erscheinungsdatum:	17.10.2021
Gewicht:	0,382 kg