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dzhafarov d reverse mathematics problems reductions proofs 2022

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Textbook in PDF format 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. Preface Acknowledgments List of Figures Introduction What is reverse mathematics? Historical remarks Considerations about coding Philosophical implications Conventions and notation 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 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 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 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 Combinatorics Ramsey's theorem 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 Ramsey's theorem for singletons Ramsey's theorem for higher exponents Homogeneity vs. limit homogeneity Partial and linear orders Equivalences and bounds Stable partial and linear orders Separations over RCA0 Variants under finer reducibilities Polarized Ramsey's theorem Rainbow Ramsey's theorem Erdős–Moser theorem The Chubb–Hirst–McNicholl tree theorem Milliken's tree theorem Thin set and free set theorems Hindman's theorem Apartness, gaps, and finite unions Towsner's simple proof Variants with bounded sums Applications of the Lovász local lemma Model theoretic and set theoretic principles Languages, theories, and models The atomic model theorem The finite intersection principle Weak weak König's lemma The reverse mathematics zoo Exercises Other areas Analysis and topology Formalizations of the real line Sequences and convergence Sets and continuous functions Sets of points Continuous functions The intermediate value theorem Closed sets and compactness Separably closed sets Uniform continuity and boundedness Topological dynamics and ergodic theory Birkhoff's recurrence theorem The Auslander–Ellis theorem and iterated Hindman's theorem Measure theory and the mean ergodic theorem Additional results in real analysis Topology, MF spaces, CSC spaces Countable, second countable spaces MF spaces Reverse mathematics of MF spaces Exercises Algebra Groups, rings, and other structures Vector spaces and bases The complexity of ideals Orderability The Nielsen–Schreier theorem Other topics Exercises Set theory and beyond Well orderings and ordinals The Sigma-1-1 separation principle Comparability of well orderings Proof of Proposition 12.1.12 Descriptive set theory Determinacy Gale–Stewart games Clopen and open determinacy Gödel's constructible universe Friedman's theorem Higher order reverse mathematics Exercises References Index

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dzhafarov d reverse mathematics problems reductions proofs 2022