LEADER 04582nam 22006495 450 001 9910585784903321 005 20251113175427.0 010 $a3-031-11367-5 024 7 $a10.1007/978-3-031-11367-3 035 $a(MiAaPQ)EBC7050334 035 $a(Au-PeEL)EBL7050334 035 $a(CKB)24281725100041 035 $a(PPN)263898709 035 $a(OCoLC)1337946026 035 $a(DE-He213)978-3-031-11367-3 035 $a(EXLCZ)9924281725100041 100 $a20220725d2022 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aReverse Mathematics $eProblems, Reductions, and Proofs /$fby Damir D. Dzhafarov, Carl Mummert 205 $a1st ed. 2022. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2022. 215 $a1 online resource (498 pages) 225 1 $aTheory and Applications of Computability, In cooperation with the Association Computability in Europe,$x2190-6203 311 08$aPrint version: Dzhafarov, Damir D. Reverse Mathematics Cham : Springer International Publishing AG,c2022 9783031113666 320 $aIncludes bibliographical references and index. 327 $a1 introduction -- Part I Computable mathematics: 2 Computability theory -- 3 Instance?solution problems -- 4 Problem reducibilities -- Part II Formalization and syntax: 5 Second order arithmetic -- 6 Induction and bounding -- 7 Forcing -- Part III Combinatorics: 8 Ramsey?s theorem -- 9 Other combinatorial principles -- Part IV Other areas: 10 Analysis and topology -- 11 Algebra -- 12 Set theory and beyond. 330 $aReverse mathematics studies the complexity of proving mathematical theorems and solving mathematical problems. Typical questions include: Can we prove this result without first proving that one? Can a computer solve this problem? A highly active part of mathematical logic and computability theory, the subject offers beautiful results as well as significant foundational insights. This text provides a modern treatment of reverse mathematics that combines computability theoretic reductions and proofs in formal arithmetic to measure the complexity of theorems and problems from all areas of mathematics. It includes detailed introductions to techniques from computable mathematics, Weihrauch style analysis, and other parts of computability that have become integral to research in the field. Topics and features: Provides a complete introduction to reverse mathematics, including necessary background from computability theory, second order arithmetic, forcing, induction, and model construction Offers a comprehensive treatment of the reverse mathematics of combinatorics, including Ramsey's theorem, Hindman's theorem, and many other results Provides central results and methods from the past two decades, appearing in book form for the first time and including preservation techniques and applications of probabilistic arguments Includes a large number of exercises of varying levels of difficulty, supplementing each chapter The text will be accessible to students with a standard first year course in mathematical logic. It will also be a useful reference for researchers in reverse mathematics, computability theory, proof theory, and related areas. Damir D. Dzhafarov is an Associate Professor of Mathematics at the University of Connecticut, CT, USA. Carl Mummert is a Professor of Computer and Information Technology at Marshall University, WV, USA. 410 0$aTheory and Applications of Computability, In cooperation with the Association Computability in Europe,$x2190-6203 606 $aComputer science$xMathematics 606 $aComputable functions 606 $aRecursion theory 606 $aLogic, Symbolic and mathematical 606 $aMathematics of Computing 606 $aComputability and Recursion Theory 606 $aMathematical Logic and Foundations 615 0$aComputer science$xMathematics. 615 0$aComputable functions. 615 0$aRecursion theory. 615 0$aLogic, Symbolic and mathematical. 615 14$aMathematics of Computing. 615 24$aComputability and Recursion Theory. 615 24$aMathematical Logic and Foundations. 676 $a511.3 676 $a511.3 700 $aDzhafarov$b Damir D.$01252240 702 $aMummert$b Carl$f1978- 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910585784903321 996 $aReverse mathematics$92999664 997 $aUNINA