LEADER 06110nam 22008655 450 001 9910299961103321 005 20200702085210.0 010 $a1-4614-8854-0 024 7 $a10.1007/978-1-4614-8854-5 035 $a(CKB)3710000000078715 035 $a(Springer)9781461488545 035 $a(MH)013884412-7 035 $a(SSID)ssj0001091877 035 $a(PQKBManifestationID)11589499 035 $a(PQKBTitleCode)TC0001091877 035 $a(PQKBWorkID)11028250 035 $a(PQKB)10307398 035 $a(DE-He213)978-1-4614-8854-5 035 $a(MiAaPQ)EBC6312495 035 $a(MiAaPQ)EBC1591883 035 $a(Au-PeEL)EBL1591883 035 $a(CaPaEBR)ebr10969152 035 $a(OCoLC)922907494 035 $a(PPN)176099875 035 $a(EXLCZ)993710000000078715 100 $a20131207d2014 u| 0 101 0 $aeng 135 $aurnn#008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aSet Theory $eWith an Introduction to Real Point Sets /$fby Abhijit Dasgupta 205 $a1st ed. 2014. 210 1$aNew York, NY :$cSpringer New York :$cImprint: Birkhäuser,$d2014. 215 $a1 online resource (XV, 444 p. 17 illus.)$conline resource 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a1-4614-8853-2 327 $a1 Preliminaries: Sets, Relations, and Functions -- Part I Dedekind: Numbers -- 2 The Dedekind?Peano Axioms -- 3 Dedekind?s Theory of the Continuum -- 4 Postscript I: What Exactly Are the Natural Numbers? -- Part II Cantor: Cardinals, Order, and Ordinals -- 5 Cardinals: Finite, Countable, and Uncountable -- 6 Cardinal Arithmetic and the Cantor Set -- 7 Orders and Order Types -- 8 Dense and Complete Orders -- 9 Well-Orders and Ordinals -- 10 Alephs, Cofinality, and the Axiom of Choice -- 11 Posets, Zorn?s Lemma, Ranks, and Trees -- 12 Postscript II: In?nitary Combinatorics -- Part III Real Point Sets -- 13 Interval Trees and Generalized Cantor Sets -- 14 Real Sets and Functions -- 15 The Heine?Borel and Baire Category Theorems -- 16 Cantor?Bendixson Analysis of Countable Closed Sets -- 17 Brouwer?s Theorem and Sierpinski?s Theorem -- 18 Borel and Analytic Sets -- 19 Postscript III: Measurability and Projective Sets -- Part IV Paradoxes and Axioms -- 20 Paradoxes and Resolutions -- 21 Zermelo?Fraenkel System and von Neumann Ordinals -- 22 Postscript IV: Landmarks of Modern Set Theory -- Appendices -- A Proofs of Uncountability of the Reals -- B Existence of Lebesgue Measure -- C List of ZF Axioms -- References -- List of Symbols and Notations -- Index. 330 $aWhat is a number? What is infinity? What is continuity? What is order? Answers to these fundamental questions obtained by late nineteenth-century mathematicians such as Dedekind and Cantor gave birth to set theory. This textbook presents classical set theory in an intuitive but concrete manner. To allow flexibility of topic selection in courses, the book is organized into four relatively independent parts with distinct mathematical flavors. Part I begins with the Dedekind?Peano axioms and ends with the construction of the real numbers. The core Cantor?Dedekind theory of cardinals, orders, and ordinals appears in Part II. Part III focuses on the real continuum. Finally, foundational issues and formal axioms are introduced in Part IV. Each part ends with a postscript chapter discussing topics beyond the scope of the main text, ranging from philosophical remarks to glimpses into landmark results of modern set theory such as the resolution of Lusin's problems on projective sets using determinacy of infinite games and large cardinals. Separating the metamathematical issues into an optional fourth part at the end makes this textbook suitable for students interested in any field of mathematics, not just for those planning to specialize in logic or foundations. There is enough material in the text for a year-long course at the upper-undergraduate level. For shorter one-semester or one-quarter courses, a variety of arrangements of topics are possible. The book will be a useful resource for both experts working in a relevant or adjacent area and beginners wanting to learn set theory via self-study. 606 $aLogic, Symbolic and mathematical 606 $aMathematical analysis 606 $aAnalysis (Mathematics) 606 $aAlgebra 606 $aTopology 606 $aDiscrete mathematics 606 $aLogic 606 $aMathematical Logic and Foundations$3https://scigraph.springernature.com/ontologies/product-market-codes/M24005 606 $aAnalysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12007 606 $aAlgebra$3https://scigraph.springernature.com/ontologies/product-market-codes/M11000 606 $aTopology$3https://scigraph.springernature.com/ontologies/product-market-codes/M28000 606 $aDiscrete Mathematics$3https://scigraph.springernature.com/ontologies/product-market-codes/M29000 606 $aLogic$3https://scigraph.springernature.com/ontologies/product-market-codes/E16000 615 0$aLogic, Symbolic and mathematical. 615 0$aMathematical analysis. 615 0$aAnalysis (Mathematics). 615 0$aAlgebra. 615 0$aTopology. 615 0$aDiscrete mathematics. 615 0$aLogic. 615 14$aMathematical Logic and Foundations. 615 24$aAnalysis. 615 24$aAlgebra. 615 24$aTopology. 615 24$aDiscrete Mathematics. 615 24$aLogic. 676 $a511.322 700 $aDasgupta$b Abhijit$4aut$4http://id.loc.gov/vocabulary/relators/aut$0721731 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910299961103321 996 $aSet theory$91410730 997 $aUNINA 999 $aThis Record contains information from the Harvard Library Bibliographic Dataset, which is provided by the Harvard Library under its Bibliographic Dataset Use Terms and includes data made available by, among others the Library of Congress