LEADER 04000nam 22005775 450 001 9911002546303321 005 20260128123612.0 010 $a9783031851063$b(electronic bk.) 010 $z9783031851056 024 7 $a10.1007/978-3-031-85106-3 035 $a(MiAaPQ)EBC32108419 035 $a(Au-PeEL)EBL32108419 035 $a(CKB)38767391700041 035 $a(DE-He213)978-3-031-85106-3 035 $a(OCoLC)1519912905 035 $a(EXLCZ)9938767391700041 100 $a20250511d2025 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aGödel's Theorems and Zermelo's Axioms $eA Firm Foundation of Mathematics /$fby Lorenz Halbeisen, Regula Krapf 205 $a2nd ed. 2025. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Birkhäuser,$d2025. 215 $a1 online resource (339 pages) 225 1 $aMathematics and Statistics Series 311 08$aPrint version: Halbeisen, Lorenz Gödel's Theorems and Zermelo's Axioms Cham : Birkhäuser Boston,c2025 9783031851056 327 $a0. 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. 330 $aThis book provides a concise and self-contained introduction to the foundations of mathematics. The first part covers the fundamental notions of mathematical logic, including logical axioms, formal proofs and the basics of model theory. Building on this, in the second and third part of the book the authors present detailed proofs of Gödel?s classical completeness and incompleteness theorems. In particular, the book includes a full proof of Gödel?s second incompleteness theorem which states that it is impossible to prove the consistency of arithmetic within its axioms. The final part is dedicated to an introduction into modern axiomatic set theory based on Zermelo?s axioms, containing also a presentation of Gödel?s constructible universe of sets. A recurring theme in the whole book consists of standard and non-standard models of several theories, such as Peano arithmetic, Presburger arithmetic and the real numbers. In addition, the corrected, revised and extended second edition now provides detailed solutions to all exercises. The book addresses undergraduate mathematics students and is suitable for a one or two semester introductory course into logic and set theory. 410 0$aMathematics and Statistics Series 606 $aLogic, Symbolic and mathematical 606 $aMathematical Logic and Foundations 606 $aTeorema de Gödel$2thub 606 $aLōgica matemātica$2thub 606 $aTeoria de conjunts$2thub 608 $aLlibres electrōnics$2thub 615 0$aLogic, Symbolic and mathematical. 615 14$aMathematical Logic and Foundations. 615 7$aTeorema de Gödel. 615 7$aLōgica matemātica 615 7$aTeoria de conjunts 676 $a511.3 700 $aHalbeisen$b Lorenz$0767629 701 $aKrapf$b Regula$01253238 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a9911002546303321 996 $aGödel's Theorems and Zermelo's Axioms$92905443 997 $aUNINA