Gödel's Theorems and Zermelo's Axioms

A Firm Foundation of Mathematics

Inbunden, Engelska, 2025

Av Lorenz Halbeisen, Regula Krapf

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This book provides a concise and self-contained introduction to the foundations of mathematics. 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.

Produktinformation

Utgivningsdatum2025-05-11
Mått155 x 235 x 26 mm
Vikt698 g
FormatInbunden
SpråkEngelska
Antal sidor342
Upplaga2
FörlagBirkhauser Verlag AG
ISBN9783031851056

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Matematikens grunder inom Naturvetenskap och teknik

0. A Framework for Metamathematics.- Part I Introduction to First-Order Logic.- 1 Syntax: The Grammar of Symbols.- 2 The Art of Proof.- 3 Semantics: Making Sense of the Symbols.- Part II Gödel’s Completeness Theorem.- 4 Maximally Consistent Extensions.- 5 The Completeness Theorem.- 6 Language Extensions by Definitions.- Part III Gödel’s Incompleteness Theorems.- 7 Countable Models of Peano Arithmetic.- 8 Arithmetic in Peano Arithmetic.- 9 Gödelisation of Peano Arithmetic.- 10 The First Incompleteness Theorem.- 11 The Second Incompleteness Theorem.- 12 Completeness of Presburger Arithmetic.- Part IV The Axiom System ZFC.- 13 The Axioms of Set Theory (ZFC).- 14 Models of Set Theory.- 15 Models and Ultraproducts.- 16 Models of Peano Arithmetic.- 17 Models of the Real Numbers.- Tautologies.- Solutions.- References.- Index.