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E-grāmata: Reverse Mathematics: Problems, Reductions, and Proofs

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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.

Recenzijas

This book takes a broad approach to reverse mathematics, emphasizing connections to computability theory and Weihrauch analysis. It will serve well as a text for courses in reverse mathematics and as a reference for researchers in the area. (Jeffry L. Hirst, zbMATH 1555.03002, 2025) 



Dzhafarov and Mummert's book provides a systematic introduction to new developments in reverse mathematics based on a computability-theoretic approach . This book is a valuable reference for students and researchers working on reverse mathematics as well as other branches of mathematical logic. (Huishan Wu, Mathematical Reviews, September, 2023)

Preface vii
Acknowledgments xi
1 Introduction
1(14)
1.1 What is reverse mathematics?
1(2)
1.2 Historical remarks
3(3)
1.3 Considerations about coding
6(2)
1.4 Philosophical implications
8(2)
1.5 Conventions and notation
10(5)
Part I Computable mathematics
2 Computability theory
15(36)
2.1 The informal idea of computability
15(2)
2.2 Primitive recursive functions
17(6)
2.2.1 Some primitive recursive functions
19(1)
2.2.2 Bounded quantification
20(2)
2.2.3 Coding sequences with primitive recursion
22(1)
2.3 Turing computability
23(3)
2.4 Three key theorems
26(5)
2.5 Computably enumerable sets and the halting problem
31(2)
2.6 The arithmetical hierarchy and Post's theorem
33(3)
2.7 Relativization and oracles
36(1)
2.8 Trees and PA degrees
37(11)
2.8.1 n1, classes
39(4)
2.8.2 Basis theorems
43(3)
2.8.3 PA degrees
46(2)
2.9 Exercises
48(3)
3 Instance-solution problems
51(26)
3.1 Problems
51(2)
3.2 V3 theorems
53(3)
3.3 Multiple problem forms
56(3)
3.4 Represented spaces
59(2)
3.5 Representing R
61(2)
3.6 Complexity
63(5)
3.7 Uniformity
68(2)
3.8 Further examples
70(4)
3.9 Exercises
74(3)
4 Problem reducibilities
77(30)
4.1 Subproblems and identity reducibility
78(3)
4.2 Computable reducibility
81(3)
4.3 Weihrauch reducibility
84(4)
4.4 Strong forms
88(3)
4.5 Multiple applications
91(4)
4.6 ω-model reducibility
95(5)
4.7 Hirschfeldt-Jockusch games
100(3)
4.8 Exercises
103(4)
Part II Formalization and syntax
5 Second order arithmetic
107(46)
5.1 Syntax and semantics
107(4)
5.2 Hierarchies of formulas
111(2)
5.2.1 Arithmetical formulas
111(2)
5.2.2 Analytical formulas
113(1)
5.3 Arithmetic
113(4)
5.3.1 First order arithmetic
113(2)
5.3.2 Second order arithmetic
115(2)
5.4 Formalization, and on ω and N
117(1)
5.5 The subsystem RCA0
118(5)
5.5.1 Δ01 comprehension
118(1)
5.5.2 Coding finite sets
119(3)
5.5.3 Formalizing computability theory
122(1)
5.6 The subsystems ACA0 and WKL0
123(6)
5.6.1 The subsystem ACA0
124(3)
5.6.2 The subsystem WKL0
127(2)
5.7 Equivalences between mathematical principles
129(3)
5.8 The subsystems πl-CA0 and ATR0
132(9)
5.8.1 The subsystem π1-CA0
132(4)
5.8.2 The subsystem ATR0
136(5)
5.9 Conservation results
141(2)
5.10 First order parts of theories
143(1)
5.11 Comparing reducibility notions
144(2)
5.12 Full second order semantics
146(2)
5.13 Exercises
148(5)
6 Induction and bounding
153(22)
6.1 Induction, bounding, and least number principles
153(5)
6.2 Finiteness, cuts, and all that
158(4)
6.3 The Kirby-Paris hierarchy
162(4)
6.4 Reverse recursion theory
166(2)
6.5 Hirst's theorem and B2Σ02
168(4)
6.6 So, why Σ01 induction?
172(2)
6.7 Exercises
174(1)
7 Forcing
175(28)
7.1 A motivating example
175(2)
7.2 Notions of forcing
177(5)
7.3 Density and genericity
182(4)
7.4 The forcing relation
186(3)
7.5 Effective forcing
189(5)
7.6 Forcing in models
194(2)
7.7 Harrington's theorem and conservation
196(3)
7.8 Exercises
199(4)
Part III Combinatorics
8 Ramsey's theorem
203(56)
8.1 Upper bounds
204(5)
8.2 Lower bounds
209(3)
8.3 Seetapun's theorem
212(4)
8.4 Stability and cohesiveness
216(7)
8.4.1 Stability
217(4)
8.4.2 Cohesiveness
221(2)
8.5 The Cholak-Jockusch-Slaman decomposition
223(8)
8.5.1 A different proof of Seetapun's theorem
225(3)
8.5.2 Other applications
228(3)
8.6 Liu's theorem
231(7)
8.6.1 Preliminaries
232(2)
8.6.2 Proof of Lemma 8.6.6
234(1)
8.6.3 Proof of Lemma 8.6.7
234(4)
8.7 The first order part of RT
238(13)
8.7.1 Two versus arbitrarily many colors
238(3)
8.7.2 Proof of Proposition 8.7.4
241(4)
8.7.3 Proof of Proposition 8.7.5
245(4)
8.7.4 What else is known?
249(2)
8.8 The SRT; vs. COH problem
251(5)
8.9 Summary: Ramsey's theorem and the "big five"
256(1)
8.10 Exercises
257(2)
9 Other combinatorial principles
259(104)
9.1 Finer results about RT
259(12)
9.1.1 Ramsey's theorem for singletons
259(6)
9.1.2 Ramsey's theorem for higher exponents
265(5)
9.1.3 Homogeneity vs. limit homogeneity
270(1)
9.2 Partial and linear orders
271(20)
9.2.1 Equivalences and bounds
272(3)
9.2.2 Stable partial and linear orders
275(4)
9.2.3 Separations over RCA0
279(7)
9.2.4 Variants under finer reducibilities
286(5)
9.3 Polarized Ramsey's theorem
291(3)
9.4 Rainbow Ramsey's theorem
294(9)
9.5 Erdos-Moser theorem
303(2)
9.6 The Chubb-Hirst-McNicholl tree theorem
305(4)
9.7 Milliken's tree theorem
309(3)
9.8 Thin set and free set theorems
312(5)
9.9 Hindman's theorem
317(21)
9.9.1 Apartness, gaps, and finite unions
319(4)
9.9.2 Towsner's simple proof
323(6)
9.9.3 Variants with bounded sums
329(7)
9.9.4 Applications of the Lovasz local lemma
336(2)
9.10 Model theoretic and set theoretic principles
338(11)
9.10.1 Languages, theories, and models
338(2)
9.10.2 The atomic model theorem
340(5)
9.10.3 The finite intersection principle
345(4)
9.11 Weak weak Konig's lemma
349(5)
9.12 The reverse mathematics zoo
354(1)
9.13 Exercises
355(8)
Part IV Other areas
10 Analysis and topology
363(38)
10.1 Formalizations of the real line
364(2)
10.2 Sequences and convergence
366(2)
10.3 Sets and continuous functions
368(3)
10.3.1 Sets of points
368(1)
10.3.2 Continuous functions
369(2)
10.4 The intermediate value theorem
371(2)
10.5 Closed sets and compactness
373(7)
10.5.1 Separably closed sets
376(3)
10.5.2 Uniform continuity and boundedness
379(1)
10.6 Topological dynamics and ergodic theory
380(8)
10.6.1 BirkhofFs recurrence theorem
381(4)
10.6.2 The Auslander-Ellis theorem and iterated Hindman's theorem
385(3)
10.6.3 Measure theory and the mean ergodic theorem
388(1)
10.7 Additional results in real analysis
388(2)
10.8 Topology, MF spaces, CSC spaces
390(7)
10.8.1 Countable, second countable spaces
390(3)
10.8.2 MF spaces
393(2)
10.8.3 Reverse mathematics of MF spaces
395(2)
10.9 Exercises
397(4)
11 Algebra
401(26)
11.1 Groups, rings, and other structures
401(2)
11.2 Vector spaces and bases
403(3)
11.3 The complexity of ideals
406(4)
11.4 Orderability
410(5)
11.5 The Nielsen-Schreier theorem
415(5)
11.6 Other topics
420(3)
11.7 Exercises
423(4)
12 Set theory and beyond
427(32)
12.1 Well orderings and ordinals
427(12)
12.1.1 The Zj separation principle
428(3)
12.1.2 Comparability of well orderings
431(4)
12.1.3 Proof of Proposition 12.1.12
435(4)
12.2 Descriptive set theory
439(4)
12.3 Determinacy
443(9)
12.3.1 Gale-Stewart games
443(2)
12.3.2 Clopen and open determinacy
445(3)
12.3.3 Godel's constructible universe
448(1)
12.3.4 Friedman's theorem
449(3)
12.4 Higher order reverse mathematics
452(5)
12.5 Exercises
457(2)
References 459(14)
Index 473
Damir D. Dzhafarov is an Associate Professor of Mathematics at the University of Connecticut. He obtained his PhD from the University of Chicago, and has held postdoctoral positions at the University of Notre Dame and the University of California, Berkeley. He has held visiting positions at the National University of Singapore and Charles University, Prague. His research focuses on the computability theoretic and reverse mathematical aspects of of combinatorics, and on the interactions of reverse mathematics with computable analysis and other areas.

Carl Mummert is a Professor of Computer and Information Technology at Marshall Univeristy. He obtained his Ph.D. from Pennsylvania State University and held postdoctoral positions at Appalachian State University and the University of Michigan.  His research has included the reverse mathematics of topology and combinatorics as well as higher order reverse mathematics.
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