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E-grāmata: Slicing The Truth: On The Computable And Reverse Mathematics Of Combinatorial Principles

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Edited by (Univ Of California, Berkeley, Usa), (The Univ Of Chicago, Usa), Edited by (Nus, S'pore), Edited by (Chinese Academy Of Sciences, China), Edited by (The Univ Of California, Berkeley, Usa), Edited by (Nus, S'pore)
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Slicing The Truth: On The Computable And Reverse Mathematics Of Combinatorial PrinciplesPalielināt
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This book is a brief and focused introduction to the reverse mathematics and computability theory of combinatorial principles, an area of research which has seen a particular surge of activity in the last few years. It provides an overview of some fundamental ideas and techniques, and enough context to make it possible for students with at least a basic knowledge of computability theory and proof theory to appreciate the exciting advances currently happening in the area, and perhaps make contributions of their own. It adopts a case-study approach, using the study of versions of Ramsey's Theorem (for colorings of tuples of natural numbers) and related principles as illustrations of various aspects of computability theoretic and reverse mathematical analysis. This book contains many exercises and open questions.
Foreword by Series Editors ix
Foreword by Volume Editors xi
Preface xiii
Acknowledgments xv
1 Setting Off: An Introduction
1(16)
1.1 A measure of motivation
2(4)
1.2 Computable mathematics
6(5)
1.3 Reverse mathematics
11(3)
1.4 An overview
14(1)
1.5 Further reading
15(2)
2 Gathering Our Tools: Basic Concepts and Notation
17(12)
2.1 Computability theory
17(4)
2.2 Computability theoretic reductions
21(2)
2.3 Forcing
23(6)
3 Finding Our Path: Konig's Lemma and Computability
29(12)
3.1 Π01 classes, basis theorems, and PA degrees
30(5)
3.2 Versions of Konig's Lemma
35(6)
4 Gauging Our Strength: Reverse Mathematics
41(28)
4.1 RCA0
44(4)
4.2 Working in RCA0
48(6)
4.3 ACA0
54(1)
4.4 WKL0
55(2)
4.5 ω-Models
57(5)
4.6 First order axioms
62(3)
4.7 Further remarks
65(4)
5 In Defense of Disarray
69(6)
6 Achieving Consensus: Ramsey's Theorem
75(50)
6.1 Three proofs of Ramsey's Theorem
76(6)
6.2 Ramsey's Theorem and the arithmetic hierarchy
82(7)
6.3 RT, ACA'0, and the Paris-Harrington Theorem
89(4)
6.4 Stability and cohesiveness
93(5)
6.5 Mathias forcing and cohesive sets
98(7)
6.6 Mathias forcing and stable colorings
105(5)
6.7 Seetapun's Theorem and its extensions
110(8)
6.8 Ramsey's Theorem and first order axioms
118(5)
6.9 Uniformity
123(2)
7 Preserving Our Power: Conservativity
125(14)
7.1 Conservativity over first order systems
127(3)
7.2 WKL0 and Π11-conservativity
130(5)
7.3 COH and r-Π12-conservativity
135(4)
8 Drawing a Map: Five Diagrams
139(4)
9 Exploring Our Surroundings: The World below RT22
143(34)
9.1 Ascending and descending sequences
143(9)
9.2 Other combinatorial principles provable from RT22
152(8)
9.2.1 Chains and antichains
152(2)
9.2.2 Tournaments
154(1)
9.2.3 Free sets and thin sets
155(2)
9.2.4 The finite intersection principle and Π01-genericity
157(3)
9.3 Atomic models and omitting types
160(17)
10 Charging Ahead: Further Topics
177(16)
10.1 The Dushnik-Miller Theorem
177(2)
10.2 Linearizing well-founded partial orders
179(4)
10.3 The world above ACA0
183(8)
10.3.1 ATR0 and Π11-CA0
183(2)
10.3.2 The extendibility of ζ and η
185(2)
10.3.3 Maximal linear extensions
187(1)
10.3.4 Kruskal's Theorem, Fraisse's Conjecture, and Jullien's Theorem
188(2)
10.3.5 Hindman's Theorem
190(1)
10.4 Still further topics, and a final exercise
191(2)
Appendix. Lagniappe: A Proof of Liu's Theorem 193(10)
Bibliography 203
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