Topics in Descriptive Set Theory

University Of North Texas, Denton TX

Investigators

Abstract

This project concerns research in set theory and descriptive set theory, particularly involving the influences of the axiom of determinacy. The axiom of determinacy is the statement that every two-player integer game is determined, that is, one of the players has a winning strategy. Although this axiom contradicts the axiom of choice, an accepted part of mathematics, it was proposed in the 1960s to be a reasonable assumption for the smallest inner model of set theory containing the real numbers. This model contains the sets of reals occurring in ordinary mathematical practice (e.g., the projective sets and beyond). The study of this model, in turn, gives direct information about the mathematical universe. The notion of a scale is a central structural concept in descriptive set theory, and the axiom of determinacy was used to develop the scale theory of the projective sets, which was later extended throughout the entire model. The scale theory by itself, however, is not sufficient to answer many questions. A more detailed inductive analysis was begun in the mid 1980s which was successful at the lower levels of the model, including the projective sets. This theory does not currently extend through the entire model, and finding such an extension remains a central goal. Recently, several new lines of investigation have opened up, relating in some way to the theory of this model. One such direction concerns investigating the connections between the existing theory of this model and the theory of ultrafilters (Shelah's p.c.f. theory). Preliminary results, which use centrally Woodin's theory of the nonstationary ideal, suggest strong collapsing principles apply as one moves from the inner model to the real universe. Exactly what is forced to collapse, and what principles govern this phenomenon, is a topic of investigation. Finding and establishing principles independent of the complete inductive analysis might also provide a framework for propagating a basic "skeleton" of the analysis through the entire model. This project attempts to advance the understanding of the mathematical universe of sets. All of mathematics takes place within this universe, and progress here is important not only foundationally, but because of the direct influence on the various branches of mathematics. It has been known for some time that strong assumptions are needed to answer many basic mathematical questions. Identifying these assumptions and exploring their consequences is a major theme in set theory. The axiom of determinacy is an important example of such an axiom. This was formulated in the 1960s but the full extent of its consequences is not known. Recent evidence suggests that it may shed light on some basic mathematical questions, such as the continuum hypothesis (the question of how many real numbers there are).