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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.
Setting Off: An Introduction
Gathering Our Tools: Basic Concepts and Notation
Finding Our Path: Kö,nig',s Lemma and Computability
Gauging Our Strength: Reverse Mathematics
In Defense of Disarray
Achieving Consensus: Ramsey',s Theorem
Preserving Our Power: Conservativity
Drawing a Map: Five Diagrams
Exploring Our Surroundings: The World Below RT22
Charging Ahead: Further Topics
Lagniappe: A Proof of Liu',s Theorem
Readership: Graduates and researchers in mathematical logic.
This book is self-contained and unified in presentation
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