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Slicing the Truth: On the Computability Theoretic and Reverse Mathematical Analysis of . . .
| Venue: | INSTITUTE FOR MATHEMATICAL SCIENCES, NATIONAL UNIVERSITY OF SINGAPORE, WORLD SCIENTIFIC |
| Citations: | 3 - 1 self |
BibTeX
@INPROCEEDINGS{Hirschfeldt_slicingthe,
author = {Denis R. Hirschfeldt},
title = {Slicing the Truth: On the Computability Theoretic and Reverse Mathematical Analysis of . . . },
booktitle = {INSTITUTE FOR MATHEMATICAL SCIENCES, NATIONAL UNIVERSITY OF SINGAPORE, WORLD SCIENTIFIC},
year = {},
publisher = {}
}
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Abstract
In this expository article, we discuss two closely related approaches to studying the relative strength of mathematical principles: computable mathematics and reverse mathematics. Drawing our examples from combinatorics and model theory, we explore a variety of phenomena and techniques in these areas. We begin with variations on König’s Lemma, and give an introduction to reverse mathematics and related parts of computability theory. We then focus on Ramsey’s Theorem as a case study in the computability theoretic and reverse mathematical analysis of com-binatorial principles. We study Ramsey’s Theorem for Pairs (RT22) in detail, focusing on fundamental tools such as stability, cohesiveness, and Mathias forcing; and on combinatorial and model theoretic consequences of RT22. We also discuss the important theme of conservativity results. In the final section, we explore several topics that reveal various aspects of computable mathematics and reverse mathematics. An appendix con-tains a proof of Liu’s recent result that RT22 does not imply Weak König’s Lemma. There are exercises and open questions throughout the article.
Keyphrases
reverse mathematical analysis computability theoretic reverse mathematics computable mathematics ramsey theorem open question various aspect important theme appendix con-tains model theory conservativity result weak ko several topic case study final section model theoretic consequence mathias forcing mathematical principle recent result related approach expository article com-putability theory fundamental tool relative strength com-binatorial principle related part
