00999nam0 22002651i 450 UON0042474020231205104842.23920130517d1961 |0itac50 baengGB|||| 1||||ˆThe ‰long revolutionRaymond WilliamsLondonChatto & Windus1961XIV, 369 p.23 cm.GRAN BRETAGNAVita artistica e culturaleSec. 20.UONC074408FIGBLondonUONL003044941.082Storia della Gran Bretagna. 1901-199921WILLIAMSRaymondUONV104679309364Chatto & WindusUONV246194650ITSOL20240220RICASIBA - SISTEMA BIBLIOTECARIO DI ATENEOUONSIUON00424740SIBA - SISTEMA BIBLIOTECARIO DI ATENEOSI Angl VII 0620 SI SI 751 5 0620 Long revolution77854UNIOR04000nam 22005775 450 991100254630332120260128123612.09783031851063(electronic bk.)978303185105610.1007/978-3-031-85106-3(MiAaPQ)EBC32108419(Au-PeEL)EBL32108419(CKB)38767391700041(DE-He213)978-3-031-85106-3(OCoLC)1519912905(EXLCZ)993876739170004120250511d2025 u| 0engurcnu||||||||txtrdacontentcrdamediacrrdacarrierGödel's Theorems and Zermelo's Axioms A Firm Foundation of Mathematics /by Lorenz Halbeisen, Regula Krapf2nd ed. 2025.Cham :Springer Nature Switzerland :Imprint: Birkhäuser,2025.1 online resource (339 pages)Mathematics and Statistics SeriesPrint version: Halbeisen, Lorenz Gödel's Theorems and Zermelo's Axioms Cham : Birkhäuser Boston,c2025 9783031851056 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.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 Zermelo’s axioms, containing also 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. In addition, the corrected, revised and extended second edition now provides detailed solutions to all exercises. The book addresses undergraduate mathematics students and is suitable for a one or two semester introductory course into logic and set theory.Mathematics and Statistics SeriesLogic, Symbolic and mathematicalMathematical Logic and FoundationsTeorema de GödelthubLògica matemàticathubTeoria de conjuntsthubLlibres electrònicsthubLogic, Symbolic and mathematical.Mathematical Logic and Foundations.Teorema de Gödel.Lògica matemàticaTeoria de conjunts511.3Halbeisen Lorenz767629Krapf Regula1253238MiAaPQMiAaPQMiAaPQ9911002546303321Gödel's Theorems and Zermelo's Axioms2905443UNINA