Gödel's Theorems and Zermelo's Axioms

Lorenz Halbeisen is Lecturer at the ETH Zürich since 2014.

“The book under review is a compact and rich manual (surprisingly rich, I would say, given its compactness), that well serves the purpose declared by the authors themselves … . The book is divided into four parts, each of which is made up of chapters. Useful exercises are offered at the end of each of them, which often nicely complete the information of the text.” (Riccardo Bruni, Mathematical Reviews, August, 2022)

• 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.