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 FirstOrder 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 selfcontained 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 nonstandard 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 

3030522784 

9783030522780 
ISBN/ISSN 
10.1007/9783030522797 
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 
