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Slicing the truthon the computable a...

Chong, C.-T. (1949-)

Slicing the truthon the computable and reverse mathematics of combinatorial principles /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Slicing the truthDenis R. Hirschfeldt ; editors, Chitat Chong ... [et al.]
其他題名:	on the computable and reverse mathematics of combinatorial principles /
作者:	Hirschfeldt, Denis Roman.
其他作者:	Chong, C.-T.
出版者:	Singapore ;World Scientific,2015.
面頁冊數:	1 online resource (xv, 214 p.)
標題:	Reverse mathematics.
電子資源:	http://www.worldscientific.com/worldscibooks/10.1142/9208#t=toc
ISBN:	9789814612623 (electronic bk.)

Slicing the truthon the computable and reverse mathematics of combinatorial principles /
Hirschfeldt, Denis Roman.

Slicing the truthon the computable and reverse mathematics of combinatorial principles /[electronic resource] :Denis R. Hirschfeldt ; editors, Chitat Chong ... [et al.] - Singapore ;World Scientific,2015. - 1 online resource (xv, 214 p.) - Lecture notes series / Institute for Mathematical Sciences, National University of Singapore ;vol. 28. - Lecture notes series (National University of Singapore. Institute for Mathematical Sciences).

Includes bibliographical references and index.

1. Setting off: An introduction. 1.1. A measure of motivation. 1.2. Computable mathematics. 1.3. Reverse mathematics. 1.4. An overview. 1.5. Further reading -- 2. Gathering our tools: Basic concepts and notation. 2.1. Computability theory. 2.2. Computability theoretic reductions. 2.3. Forcing -- 3. Finding our path: Konig's lemma and computability. 3.1. II[symbol] classes, basis theorems, and PA degrees. 3.2. Versions of Konig's lemma -- 4. Gauging our strength: Reverse mathematics. 4.1. RCA[symbol]. 4.2. Working in RCA[symbol]. 4.3. ACA[symbol]. 4.4. WKL[symbol]. 4.5. [symbol]-models. 4.6. First order axioms. 4.7. Further remarks -- 5. In defense of disarray -- 6. Achieving consensus: Ramsey's theorem. 6.1. Three proofs of Ramsey's theorem. 6.2. Ramsey's theorem and the arithmetic hierarchy. 6.3. RT, ACA[symbol], and the Paris-Harrington theorem. 6.4. Stability and cohesiveness. 6.5. Mathias forcing and cohesive sets. 6.6. Mathias forcing and stable colorings. 6.7. Seetapun's theorem and its extensions. 6.8. Ramsey's theorem and first order axioms. 6.9. Uniformity -- 7. Preserving our power: Conservativity. 7.1. Conservativity over first order systems. 7.2. WKL[symbol] and II[symbol]-conservativity. 7.3. COH and r-II[symbol]-conservativity -- 8. Drawing a map: Five diagrams -- 9. Exploring our surroundings: The world below RT[symbol]. 9.1. Ascending and descending sequences. 9.2. Other combinatorial principles provable from RT[symbol]. 9.3. Atomic models and omitting types -- 10. Charging ahead: Further topics. 10.1. The Dushnik-Miller theorem. 10.2. Linearizing well-founded partial orders. 10.3. The world above ACA[symbol]. 10.4. Still further topics, and a final exercise.

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.

ISBN: 9789814612623 (electronic bk.)Subjects--Topical Terms:

717078
Reverse mathematics.

LC Class. No.: QA9.25 / .H57 2015eb

Dewey Class. No.: 511/.6

Slicing the truthon the computable and reverse mathematics of combinatorial principles /
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100 1 $a Hirschfeldt, Denis Roman. $3 717076
245 1 0 $a Slicing the truth $h [electronic resource] : $b on the computable and reverse mathematics of combinatorial principles / $c Denis R. Hirschfeldt ; editors, Chitat Chong ... [et al.]
260 $a Singapore ; $a Hackensack, NJ : $b World Scientific, $c 2015.
300 $a 1 online resource (xv, 214 p.)
490 1 $a Lecture notes series / Institute for Mathematical Sciences, National University of Singapore ; $v vol. 28
504 $a Includes bibliographical references and index.
505 0 $a 1. Setting off: An introduction. 1.1. A measure of motivation. 1.2. Computable mathematics. 1.3. Reverse mathematics. 1.4. An overview. 1.5. Further reading -- 2. Gathering our tools: Basic concepts and notation. 2.1. Computability theory. 2.2. Computability theoretic reductions. 2.3. Forcing -- 3. Finding our path: Konig's lemma and computability. 3.1. II[symbol] classes, basis theorems, and PA degrees. 3.2. Versions of Konig's lemma -- 4. Gauging our strength: Reverse mathematics. 4.1. RCA[symbol]. 4.2. Working in RCA[symbol]. 4.3. ACA[symbol]. 4.4. WKL[symbol]. 4.5. [symbol]-models. 4.6. First order axioms. 4.7. Further remarks -- 5. In defense of disarray -- 6. Achieving consensus: Ramsey's theorem. 6.1. Three proofs of Ramsey's theorem. 6.2. Ramsey's theorem and the arithmetic hierarchy. 6.3. RT, ACA[symbol], and the Paris-Harrington theorem. 6.4. Stability and cohesiveness. 6.5. Mathias forcing and cohesive sets. 6.6. Mathias forcing and stable colorings. 6.7. Seetapun's theorem and its extensions. 6.8. Ramsey's theorem and first order axioms. 6.9. Uniformity -- 7. Preserving our power: Conservativity. 7.1. Conservativity over first order systems. 7.2. WKL[symbol] and II[symbol]-conservativity. 7.3. COH and r-II[symbol]-conservativity -- 8. Drawing a map: Five diagrams -- 9. Exploring our surroundings: The world below RT[symbol]. 9.1. Ascending and descending sequences. 9.2. Other combinatorial principles provable from RT[symbol]. 9.3. Atomic models and omitting types -- 10. Charging ahead: Further topics. 10.1. The Dushnik-Miller theorem. 10.2. Linearizing well-founded partial orders. 10.3. The world above ACA[symbol]. 10.4. Still further topics, and a final exercise.
520 $a 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.
588 $a Description based on print version record.
650 0 $a Reverse mathematics. $3 717078
650 0 $a Combinatorial analysis. $3 182280
700 1 $a Chong, C.-T. $q (Chi-Tat), $d 1949- $3 665956
830 0 $a Lecture notes series (National University of Singapore. Institute for Mathematical Sciences) $3 717077
856 4 0 $u http://www.worldscientific.com/worldscibooks/10.1142/9208#t=toc