This project is to study natural structures in set theory for the deeper understanding of models of set theory and various objects inside and outside set theory. A natural structure is informally defined as a structure whose behavior is largely independent of the choice made by Axiom of Choice. There are some traditional natural structures, but new natural structures were introduced in the past few decades, particularly in PCF theory and minimal walk theory. These new natural structures play a central role in the recent development of set theory. Natural ideals form an important class of natural structures. The PI has been successful in investigating large cardinal properties of a class of natural ideals, called club guessing ideals. In this project, he studies this material further and also extends the research to other natural ideals, such as the weakly compact ideal, the ineffable ideal, and the non-diamond ideal. The PI also looks for more applications of natural structures to general topology, linearly ordered sets, and other fields of mathematics.
Set theory is largely accepted as foundation of mathematics. It means that almost every field of mathematics can be expressed and scrutinized in terms of set theory. It is known that there are a wide variety of models of set theory. Until few decades ago, it was believed that very few non-trivial properties were common in all models of set theory and additional assumptions were required to conduct deeper investigation. However, the recent research, particularly in PCF theory and minimal walk theory, revealed many unexpected and profound results that hold in every model of set theory. Essential in these results is the use of natural structures. Their uniform behavior in all models help us prove theorems without any additional assumption. Moreover, the set-theoretic approach via natural structures has been successfully applied to other fields of mathematics. The PI further investigates natural structures for the deeper understanding of models of set theory. He also continues his effort to apply them to other fields of mathematics, such as general topology, model theory, and linearly ordered sets.
