Reverse Mathematics of Ramsey\u27s Theorem

Reverse mathematics aims to determine which set theoretic axioms are necessary to prove the theorems outside of the set theory. Since the 1970’s, there has been an interest in applying reverse mathematics to study combinatorial principles like Ramsey’s theorem to analyze its strength and relation to other theorems. Ramsey’s theorem for pairs states that for any infinite complete graph with a finite coloring on edges, there is an infinite subset of nodes all of whose edges share one color. In this thesis, we introduce the fundamental terminology and techniques for reverse mathematics, and demonstrate their use in proving Kőnig\u27s lemma and Ramsey\u27s theorem over RCA0

Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination

This paper is intended to provide an introduction to cut elimination which is accessible to a broad mathematical audience. Gentzen's cut elimination theorem is not as well known as it deserves to be, and it is tied to a lot of interesting mathematical structure. In particular we try to indicate some dynamical and combinatorial aspects of cut elimination, as well as its connections to complexity theory. We discuss two concrete examples where one can see the structure of short proofs with cuts, one concerning feasible numbers and the other concerning "bounded mean oscillation" from real analysis

逆数学と可算代数系

Tohoku University山崎武課

Coherent approximation of distributed expert assessments

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2011.Cataloged from PDF version of thesis.Includes bibliographical references (p. 157-168).Expert judgments of probability and expectation play an integral role in many systems. Financial markets, public policy, medical diagnostics and more rely on the ability of informed experts (both human and machine) to make educated assessments of the likelihood of various outcomes. Experts however are not immune to errors in judgment (due to bias, quantization effects, finite information or many other factors). One way to compensate for errors in individual judgments is to elicit estimates from multiple experts and then fuse the estimates together. If the experts act sufficiently independently to form their assessments, it is reasonable to assume that individual errors in judgment can be negated by pooling the experts' opinions. Determining when experts' opinions are in error is not always a simple matter. However, one common way in which experts' opinions may be seen to be in error is through inconsistency with the known underlying structure of the space of events. Not only is structure useful in identifying expert error, it should also be taken into account when designing algorithms to approximate or fuse conflicting expert assessments. This thesis generalizes previously proposed constrained optimization methods for fusing expert assessments of uncertain events and quantities. The major development consists of a set of information geometric tools for reconciling assessments that are inconsistent with the assumed structure of the space of events. This work was sponsored by the U.S. Air Force under Air Force Contract FA8721- 05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States Government.by Peter B. Jones.Ph.D