Large Cardinals and the Methodology of Mathematics

University Of Florida, Gainesville FL

Investigators

Abstract

The principal investigator studies the impact of large cardinals on the methodology of mathematics. All results stated below use large cardinal assumptions, and some such assumptions are necessary. One line of research for the funded project is the investigation of classes of objects with the same properties, so-called terminal classes. A typical result states that the Ramsey ultrafilters are a terminal class: roughly, they share all properties invariant under the Rudin-Keisler equivalence of filters. Open questions under investigation involve finding further such terminal classes, and more importantly, the quantification of theproliferation of such classes throughout mathematics. Another line of research pursued is the construction of models in which the behavior of cardinal invariants of the continuum is optimal. A typical result is that the Miller model is the optimal way of increasing the dominating number d: roughly, all projectively defined invariants which are consistently less than d, are less than d in this model. The dual result states that there is an optimal Pmax model in which the bounding number is small. Open problems involve finding further cardinal invariants which have canonical models associated with them. More challenging is the investigation of the notion of duality mentioned above, and the investigation of the limits of the method of forcing with simply definable partial orders. Set theorists have for a long time studied certain additional axioms for mathematics, called large cardinal axioms. While they are largely irrelevant for solving specific problems in most traditional fields of mathematics, they do have a strong influence on the methodology used. Typically the large cardinal axioms allow the mathematician to select an optimal approach to answering a question only by considering the syntactical form of the question. Frequently this information can serve to discover the core of a seemingly complex problem. Three examples are in order. As the first example, it has been known for twenty years that sets of reals with simple definitions are well behaved from the point of view of mathematical analysis. Second, the PI has identified several classes of objects in mathematical practice that are "terminal": all objects in such a class have the same properties. Such classes have great methodological significance, and the PI plans to isolate more of them. In still another development, the PI found that for certain mathematically important classes of theories, there is an optimal approach to answering the question of whether the theories contain no contradictions. Again, the PI continues to isolate further such classes. Generally, the funded project serves to show that such seemingly esoteric hypotheses as large cardinal axioms have direct impact on mathematical practice, thus promoting the interaction between logic, set theory and other branches of mathematics.