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Author Halbeisen, Lorenz,
Title Godel's theorems and Zermelo's axioms : a firm foundation of mathematics / Lorenz Halbeisen, Regula Krapf.
Imprint Cham, Switzerland : Birkhauser, [2020]

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Author Halbeisen, Lorenz,
Subject Godel's theorem.
Set theory.
Logic, Symbolic and mathematical.
Alt Name Krapf, Regula,
Description 1 online resource (x, 236 pages)
Bibliography Note Includes bibliographical references and index.
Contents 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 Godel's Completeness Theorem -- Maximally Consistent Extensions -- Models of Countable Theories -- The Completeness Theorem -- Language Extensions by Definitions -- Part III Godel's Incompleteness Theorems -- Models of Peano Arithmetic and Consequences for Logic -- Arithmetic in Peano Arithmetic -- Godelisation 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.
Summary 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 Godel's classical completeness and incompleteness theorems. In particular, the book includes a full proof of Godel'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 Godel'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.
Note Description based on online resource; title from digital title page (viewed on December 31, 2020).
ISBN 9783030522797 electronic book
3030522792 electronic book
ISBN/ISSN 10.1007/978-3-030-52279-7
OCLC # 1204143360
Link Springer Nature eBook
Additional Format Print version: Halbeisen, Lorenz Godel's Theorems and Zermelo's Axioms : A Firm Foundation of Mathematics Cham : Springer International Publishing AG,c2020 9783030522780

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