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|a 1337523666
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|a 10.1007/978-3-031-11367-3
|2 doi
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|a (OCoLC)1337946026
|z (OCoLC)1337523666
|z (OCoLC)1374608939
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|a HCDD
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|a Dzhafarov, Damir D.
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|a Reverse mathematics :
|b problems, reductions, and proofs /
|c Damir D. Dzhafarov, Carl Mummert.
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|a Cham :
|b Springer,
|c 2022.
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| 300 |
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|a 1 online resource (498 pages)
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| 336 |
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|a text
|b txt
|2 rdacontent
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|a computer
|b c
|2 rdamedia
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|a online resource
|b cr
|2 rdacarrier
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|a text file
|2 rdaft
|0 http://rdaregistry.info/termList/fileType/1002
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| 490 |
1 |
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|a Theory and applications of computability
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0 |
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|a Intro -- Preface -- Acknowledgments -- Contents -- List of Figures -- Introduction -- What is reverse mathematics? -- Historical remarks -- Considerations about coding -- Philosophical implications -- Conventions and notation -- Part I Computable mathematics -- Computability theory -- The informal idea of computability -- Primitive recursive functions -- Some primitive recursive functions -- Bounded quantification -- Coding sequences with primitive recursion -- Turing computability -- Three key theorems -- Computably enumerable sets and the halting problem
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|a The arithmetical hierarchy and Post's theorem -- Relativization and oracles -- Trees and PA degrees -- Pi-0-1 classes -- Basis theorems -- PA degrees -- Exercises -- Instance-solution problems -- Problems -- Forall/exists theorems -- Multiple problem forms -- Represented spaces -- Representing R -- Complexity -- Uniformity -- Further examples -- Exercises -- Problem reducibilities -- Subproblems and identity reducibility -- Computable reducibility -- Weihrauch reducibility -- Strong forms -- Multiple applications -- Omega model reducibility -- Hirschfeldt-Jockusch games -- Exercises
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|a Part II Formalization and syntax -- Second order arithmetic -- Syntax and semantics -- Hierarchies of formulas -- Arithmetical formulas -- Analytical formulas -- Arithmetic -- First order arithmetic -- Second order arithmetic -- Formalization -- The subsystem RCAo -- Delta-0-1 comprehension -- Coding finite sets -- Formalizing computability theory -- The subsystems ACAo and WKLo -- The subsystem ACA0 -- The subsystem WKL0 -- Equivalences between mathematical principles -- The subsystems P11-CAo and ATRo -- The subsystem Pi-1-1-CA0 -- The subsystem ATR0 -- Conservation results
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|a First order parts of theories -- Comparing reducibility notions -- Full second order semantics -- Exercises -- Induction and bounding -- Induction, bounding, and least number principles -- Finiteness, cuts, and all that -- The Kirby-Paris hierarchy -- Reverse recursion theory -- Hirst's theorem and B-Sigma02 -- So, why Sigma-01 induction? -- Exercises -- Forcing -- A motivating example -- Notions of forcing -- Density and genericity -- The forcing relation -- Effective forcing -- Forcing in models -- Harrington's theorem and conservation -- Exercises -- Part III Combinatorics -- Ramsey's theorem
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|a Upper bounds -- Lower bounds -- Seetapun's theorem -- Stability and cohesiveness -- Stability -- Cohesiveness -- The Cholak-Jockusch-Slaman decomposition -- A different proof of Seetapun's theorem -- Other applications -- Liu's theorem -- Preliminaries -- Proof of Lemma 8.6.6 -- Proof of Lemma 8.6.7 -- The first order part of RT -- Two versus arbitrarily many colors -- Proof of Proposition 8.7.4 -- Proof of Proposition 8.7.5 -- What else is known? -- The SRT22 vs. COH problem -- Summary: Ramsey's theorem and the ``big five'' -- Exercises -- Other combinatorial principles -- Finer results about RT
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| 500 |
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|a Ramsey's theorem for singletons
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|a Reverse 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.
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|a Includes bibliographical references and index.
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|a Online resource; title from PDF title page (SpringerLink, viewed August 8, 2022).
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| 650 |
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|a Reverse mathematics.
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| 650 |
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|a Reverse mathematics
|2 fast
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| 700 |
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|a Mummert, Carl,
|d 1978-
|1 https://id.oclc.org/worldcat/entity/E39PBJgRMRK68PVmpxbmF3RxjC
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| 776 |
0 |
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|i Print version:
|a Dzhafarov, Damir D.
|t Reverse Mathematics.
|d Cham : Springer International Publishing AG, ©2022
|z 9783031113666
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| 830 |
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|a Theory and applications of computability.
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| 856 |
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|u https://holycross.idm.oclc.org/login?auth=cas&url=https://link.springer.com/10.1007/978-3-031-11367-3
|y Click for online access
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|a SPRING-COMP2022
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| 994 |
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|a 92
|b HCD
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