New Directions in Reverse Mathematics and Applied Computability Theory

University Of Connecticut, Storrs CT

Investigators

Abstract

Mathematics today benefits from having "firm foundations", by which we usually mean a system of axioms sufficient to prove the various theorems we care about. But given a particular theorem, can we specify precisely which axioms are needed to derive it? This is a natural question, and also an ancient one: over 2000 years ago, the Greek mathematicians were asking it about Euclid's geometry. Reverse mathematics is an area of mathematical logic that offers a modern approach to this kind of question, by classifying mathematical theorems according to their logical strength. This offers a deeper insight into the fundamental ideas and methods needed to prove a given theorem. More precisely, reverse mathematics provides a framework in which to compare and contrast results from disparate areas of mathematics, which helps elucidate the underpinnings of various branches of the mathematical sciences, and thereby leads to a better understanding of mathematics and its applications. A striking fact repeatedly demonstrated in this area is that the vast majority of mathematical propositions can be classified into one of a small number of categories. But for some very important and fundamental theorems this is not the case. Dzhafarov's research focuses on theorems of this ?irregular" type, including Ramsey's theorem, various equivalents of the axiom of choice, and principles arising from certain problems in cognitive science. In this project, Dzhafarov will work to achieve a greater understanding of the complexities of these "irregular" theorems, to find new examples of such theorems from previously unexplored areas of mathematics, and to apply the reverse mathematics analysis to questions from outside of mathematics. This will be facilitated by the application of methods from computability theory and proof theory, and by the addition of ideas from various collaborations across a number of areas of pure and applied mathematics, as well as interactions with members of the multidisciplinary University of Connecticut logic group.