Basic information on set theory, and my research in particular

My research involves set theory and related fields, such as set-theoretic topology and measure theory, set-theoretic model theory and mathematical logic in general. Set theory as a discipline of mathematical logic is deeply connected with other branches of mathematics, and such connections are what I mean by applications. Typical fields in which set theory has been applied are set-theoretic topology, Boolean algebras, measure theory, group theory, to mention only a few. This ever-evolving interaction between set-theory per se and its applications, is a two-way relationship: discovering new set-theoretic properties has often been inspired by a question from another field of mathematics, and vice versa, a new theory inside of set theory has often been tested by finding applications of that theory to questions from the outside.

Among the various set-theoretic disciplines, I have been involved with proper forcing, pcf, set-theoretic topology and measure theory, set-theoretic model theory, infinite combinatorics and large cardinals forcing.

At one time, set theory was regarded as an encapturing universal theory behind mathematics, whose role is to axiomatize the way we think in mathematics. We have known for many years that a perfection in this direction is provably impossible. The fundamental results that people had in this direction have opened the way for many new branches in set-theory, such as independence results and fine structure theory. There are various views of what axiom system one should take as a basis of discourse when discussing foundational aspects of mathematics, but it seems safe to say that the Zermelo-Fraenkel+ Choice (ZFC) is the most popular system. My own view is that it is practical to take ZFC as our working theory, but to learn about it we need both the methods used to prove ZFC theorems, such as pcf theory, and the methods used to show us that no ZFC result is possible, that is the independence proofs. Many questions which can be formulated in simple combinatorial terms, for example if cardinal arithmetic is trivial, need large cardinals to become interesting. That is why modern set theory is deeply hinged on the study of large cardinals, in addition to the study of ZFC. Another important topic in set theory are determinacy axioms and inner models, both of which are closely related to the study of large cardinals.

There is, of course, much more to set theory than fits on this page, and more than I know about. This is a field which will not leave disappointed those who look for a continuous challenge and beauty. My immediate goals in the subject are to work on the topic of cardinal spectra in its various instances, both combinatorial and forcingwise, as well as to continue work on universality, a subject which have interested me for a while. And of course, to continue learning.