The department members who do research in Set Theory are Liljana Babinkostova , Stefan Geschke, Randall Holmes, Justin Moore and Marion Scheepers.
The set theory and logic seminar meets every 1-2 weeks to discuss our ongoing research and the research of others in our field. Below is a description of our individual interests.
Randall Holmes has parallel research programs in computer assisted reasoning and in set theories and other logical systems related to Quine's "New Foundations" (NF), especially Jensen's set theory NFU. In computer assisted reasoning, Holmes has developed the Watson theorem prover, an interactive equational theorem prover which uses a stratified lambda-calculus related to NFU as its higher order logic, and also embodies unusual approaches to rewriting and to the use of expressions defined by cases. Other current projects in this area include the development of a complete polymorphic type checker for the ramified type theory of Russell and Whitehead's Principia, and a sequent calculus prover for NFU based on a system defined by Marcel Crabbe. In the pure set theory area, recent investigations have been in the area of strong axioms of infinity suitable to be adjoined to NFU, and their relationship with strong axioms of infinity usually adjoined to ZFC. He recently wrote a paper on the philosophical question as to whether it is possible to present NFU as an independently motivated foundation for mathematics. Holmes is in the planning stages of writing a monograph on the consistent subsystems of NF.
Justin Moore's focus is on Ramsey theory and combinatorics as it relates to the real numbers and to the first uncountable cardinal. He studies forcing axioms such as Martin's Axiom and the Proper Forcing Axiom and how they relate to the properties of the continuum and uncountable structures. One of the goals is to gain a better understanding of the impact combinatorial statements have on the size of the continuum and in particular in its relationship to the second uncountable cardinal. Another is to understand the relationships various chain conditions --- such as Suslin's countable chain condition and Knaster's condition --- have to one another and the strengths of the axioms which equate them. Yet another is an analysis of random graphs on uncountable vertex sets using Martin's axiom and other forcing axioms. This has applications in general topology and infinitary combinatorics.
Marion Scheepers' set theory research focuses on combinatorics, game theory and Ramsey theory. The area of Selection Principles in Mathematics is an outgrowth of his research.