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.
{"title":"Slicing the Truth - On the Computable and Reverse Mathematics of Combinatorial Principles","authors":"D. Hirschfeldt","doi":"10.1142/9208","DOIUrl":"https://doi.org/10.1142/9208","url":null,"abstract":"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.","PeriodicalId":405067,"journal":{"name":"Lecture Notes Series / Institute for Mathematical Sciences / National University of Singapore","volume":"90 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133964679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Baire Category Theorem and the Baire Category Numbers Coding into the Reals Descriptive Set-Theoretic Consequences Measure-Theoretic Consequences Variations on the Souslin Hypothesis The S- and L-Space Problems The Side-Condition Method Ideal Dichotomies Coherent and Lipschitz Trees Applications to the S-Space Problem and the Von Neumann Problem Biorthogonal Systems Structure of Compact Spaces Ramsey Theory on Ordinals Five Cofinal Types Five Linear Orderings mm and Cardinal Arithmetic Reflection Principles Appendices: Basic Notions Preserving Stationary Sets Historical and Other Comments.
{"title":"Notes on Forcing Axioms","authors":"S. Todorcevic","doi":"10.1142/9013","DOIUrl":"https://doi.org/10.1142/9013","url":null,"abstract":"The Baire Category Theorem and the Baire Category Numbers Coding into the Reals Descriptive Set-Theoretic Consequences Measure-Theoretic Consequences Variations on the Souslin Hypothesis The S- and L-Space Problems The Side-Condition Method Ideal Dichotomies Coherent and Lipschitz Trees Applications to the S-Space Problem and the Von Neumann Problem Biorthogonal Systems Structure of Compact Spaces Ramsey Theory on Ordinals Five Cofinal Types Five Linear Orderings mm and Cardinal Arithmetic Reflection Principles Appendices: Basic Notions Preserving Stationary Sets Historical and Other Comments.","PeriodicalId":405067,"journal":{"name":"Lecture Notes Series / Institute for Mathematical Sciences / National University of Singapore","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2013-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123656310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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