LEADER 05217nam 22006375 450 001 9911007454503321 005 20260126133550.0 010 $a3-031-91752-9 024 7 $a10.1007/978-3-031-91752-3 035 $a(CKB)39124547700041 035 $a(DE-He213)978-3-031-91752-3 035 $a(MiAaPQ)EBC32145018 035 $a(Au-PeEL)EBL32145018 035 $a(OCoLC)1521955519 035 $a(EXLCZ)9939124547700041 100 $a20250527d2025 u| 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aCombinatorial Set Theory $eWith a Gentle Introduction to Forcing /$fby Lorenz J. Halbeisen 205 $a3rd ed. 2025. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Springer,$d2025. 215 $a1 online resource (XVII, 616 p.) 225 1 $aSpringer Monographs in Mathematics,$x2196-9922 311 08$a3-031-91751-0 327 $aPart I: Preliminary -- 1 The Setting -- 2 First-Order Logic in a Nutshell -- 3 Axioms of Set Theory -- Part II: Topics in Combinatorial Set Theory -- 4 Overture: Ramsey's Theorem -- 5 Cardinal Relations in ZF Only -- 6 Forms of Choice -- 7 How to Make Two Balls from One -- 8 Models of Set Theory with Atoms -- 9 Thirteen Cardinals and Their Relations -- 10 The Shattering Number Revisited -- 11 Happy Families and Their Relatives -- 12 Coda: A Dual Form of Ramsey?s Theorem -- Part III: From Martin?s Axiom to Cohen?s Forcing -- 13 The Idea of Forcing -- 14 Martin's Axiom -- 15 The Notion of Forcing -- 16 Proving Unprovability -- 17 Models in Which AC Fails -- 18 Combining Forcing Notions -- 19 Models in Which p=c -- 20 Suslin?s Problem -- Part IV: Combinatorics of Forcing Extensions -- 21 Properties of Forcing Extensions -- 22 Cohen Forcing Revisited -- 23 Sacks Forcing -- 24 Silver-Like Forcing Notions -- 25 Miller Forcing -- 26 Mathias Forcing -- 27 Laver Forcing -- 28 How Many Ramsey Ultrafilters Exist? -- 29 Suite. 330 $aThis book, now in a revised and extended third edition, provides a comprehensive and accessible introduction to modern axiomatic set theory. After an overview of basic notions in combinatorics and first-order logic, and discussing in great detail the axioms of set theory, the author outlines in the second part the main topics of classical set theory, including Ramsey theory and the axiom of choice. As an application of the axiom of choice, a complete proof of Robinson's construction for doubling a ball by dividing it into only five parts is given. For the new edition, the chapter on permutation models has been extended, and recent results in set theory without the axiom of choice and about cardinal characteristics have been added. The third part explains the sophisticated technique of forcing from scratch, now including more details about iterated forcing. The technique is then used to show that certain statements are neither provable nor disprovable from the axioms of set theory. In particular, it is shown that both Martin's Axiom and Suslin's Hypothesis are independent of the axioms of set theory. The final part, with a new chapter on Laver forcing, is mainly concerned with consistency results obtained by iterations of forcing notions such as Cohen forcing, Sacks forcing, and Mathias forcing. The part begins with an extended chapter on countable support iterations of proper forcing notions, now also including proofs of some preservation theorems such as preservation of properness and of certain ultrafilters. In the following chapters, various consistency results concerning possible relations between cardinal characteristics and the existence of Ramsey ultrafilters are presented. For example, a detailed proof of Shelah?s astonishing construction of a model with finitely many Ramsey ultrafilters is given. Written for graduate students in axiomatic set theory, Combinatorial Set Theory will appeal to all researchers interested in the foundations of mathematics. With extensive reference lists, historical remarks, and related results at the end of the chapters, this book is also suitable for self-study. 410 0$aSpringer Monographs in Mathematics,$x2196-9922 606 $aLogic, Symbolic and mathematical 606 $aSet theory 606 $aDiscrete mathematics 606 $aMathematical Logic and Foundations 606 $aSet Theory 606 $aDiscrete Mathematics 606 $aTeoria combinatòria de conjunts$2thub 606 $aForcing (Teoria de models)$2thub 608 $aLlibres electrònics$2thub 615 0$aLogic, Symbolic and mathematical. 615 0$aSet theory. 615 0$aDiscrete mathematics. 615 14$aMathematical Logic and Foundations. 615 24$aSet Theory. 615 24$aDiscrete Mathematics. 615 7$aTeoria combinatòria de conjunts 615 7$aForcing (Teoria de models) 676 $a511.3 700 $aHalbeisen$b Lorenz J$4aut$4http://id.loc.gov/vocabulary/relators/aut$0767629 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9911007454503321 996 $aCombinatorial Set Theory$91563063 997 $aUNINA