Research in the Foundations of Mathematics

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

Award: DMS-0245349 Principal Investigator: Harvey M. Friedman Friedman proposes to continue his efforts into extending the scope of the incompleteness phenomena. Under prior NSF support, Friedman has discovered a new mathematical theory which seeks to analyze the Boolean relations that hold between sets and their images under functions of several variables. This new Boolean relation theory seeks to analyze statements of the form "for all functions of a certain kind, there exist sets of a certain kind, such that a given Boolean relation holds among the sets and their images under the functions". Under prior NSF support, Friedman discovered a "singular" statement of a particularly simple form in Boolean relation theory that can be proved only by going beyond the usual axioms for mathematics. Friedman has considered all 6561 statements of the same simple form and showed that all can be proved or refuted using weak axioms, with the sole exception (up to symmetry) of the "singular" statement. Friedman proposes to develop Boolean relation theory in several directions, including expanding the set of 6561 statements, and shifting to many diverse mathematical contexts. By the early part of the 20th century, the standard axioms and rules of mathematics had been established - the so called Zermelo Frankel axioms of set theory (ZFC). In the 1930's, Kurt Godel stunned the mathematical world with his incompleteness theorems that showed that any systematization such as ZFC is incomplete. I.e., there will always remain sentences that can neither be proved nor refuted within that systematization. This is normally referred to as the incompleteness phenomenom. Godel's original examples of statements of unprovable and unrefutable in ZFC were very far removed from the usual considerations of mathematicians. Through a series of developments, starting with later work of Godel and Cohen, various specialists in set theory, work of Friedman recognized by the NSF Alan T. Waterman Award in 1984, and more recent work of Friedman, a body of such examples has been built up that are of increasing relevance to normal mathematical considerations. Recent work of Friedman along these lines gives new reasons for rethinking and extending the usual ZFC axioms for mathematics. To test the broader impact of the research, Friedman actively seeks and obtains regular feedback on the "naturalness" and "normality" of the various examples from the wider mathematical community. Friedman seeks to widen this broader impact.