The scientific tasks of INFTY are the following:
A. The structure of definable subsets of the continuum. This task deals with the study of definable subsets of the real line (and general Polish spaces), its applications to various parts of mathematics (real and functional analysis, harmonic analysis, ergodic theory, dynamical systems, operator algebras), and its links with set theory (determinacy of games, large cardinals, and inner models). The main themes are:
- Hierarchies of definable sets: Wadge, Borel, and projective hierarchies.
- Determinacy of games and regularity properties of definable sets of real numbers.
- Complexity of "natural sets" in analysis.
- Classification of definable structures on the real numbers: linear and partial orderings, Polish groups actions, descriptive dynamics, and definable equivalence relations.
B. Infinite combinatorics. The tools of modern combinatorial set theory, such as combinatorial principles, partition calculus, infinite trees, ultrapowers, forcing axioms, and large cardinal axioms, have been very successful in resolving problems in areas such as general topology, abstract functional analysis, measure theory, and algebra. Some of the main topics we will pursue are:
- Forcing axioms. Applications of Martin's Axiom, the Proper Forcing Axiom, and Martin's Maximum to general topology, measure theory, and algebra. A recent success has been the solution of the L-space problem.
- Iterated forcing techniques. We are interested in the development of forcing techniques with the continuum bigger than aleph_2.
- Infinitary languages and classification of uncountable structures.
- Cardinal arithmetic and large cardinals: The aim is to prove absolute upper bounds in cardinal arithmetic. A major open question is whether the set of possible cofinalities of a countable set of regular cardinals is necessarily countable. Shelah's theory of possible cofinalities (pcf) provides a new point of view and a powerful tool for applications in general topology, infinite permutation groups, Boolean algebras, etc.
C. Inner models of large cardinals and aspects of determinacy. The main work is on the Inner Model Program, whose goal is to associate to each large cardinal axiom a minimal canonical model whose structure can be analyzed in detail. These models provide evidence for the consistency of large cardinal axioms. Our investigations concentrate in the following areas.
- Analysis of the fine structure of these models, and study of general iterability criteria, in particular a unifying approach to iterability past a Woodin cardinal.
- Construction of higher core models (i.e., admitting "many" Woodin cardinals).
- Combinatorial principles in inner models and their transfer to the surrounding universe. -Aspects of determinacy: The goal is to explain the useful but ad hoc determinacy hypotheses and their numerous mathematically intriguing consequences by deriving them from strong axioms of infinity possessing intrinsic evidence.
D. Applications of set theory to Banach spaces, algebra, topology and measure theory. Some examples of applications are the following:
- Infinite combinatorics, especially Ramsey combinatorics, in Banach space theory, and also in the study of partition regular systems of linear equations, generalizing Van der Waerden's classical result on arithmetical progressions.
- Descriptive set-theoretic phenomena in topology, measure theory, and ergodic theory. -Forcing constructions of Boolean algebras.
- Set-theoretic constructions of special subsets of the real line.
- Combinatorial methods in topology.
- Large cardinals in category theory and homological algebra.
- Characterization of module-theoretic problems by cardinal properties. Construction of complicated modules over commutative rings using infinite combinatorics.
- Independence results in algebra, analysis, and topology. E.g., consistency and independence of relatives of the Whitehead problem.
E. Constructive set theory and new models of computation. Some of the main topics are the following:
- The development of constructive/computational mathematics within constructive set theory, and also the metamathematics of constructive set theory.
- Sheaf and realisability models for Aczel's CZF, connections with type theory, computability, categorical logic (algebraic set theory), intuitionistic metamathematics.
- Computation on generalized machines with tapes or registers of infinite ordinal length.
- Formal mathematics; proof-checking systems with natural language interfaces, in collaboration with linguistics.
- Applications of set theory to the formal semantics of natural language, proof-checking systems with natural language interfaces, fuzziness in set theory and natural language.
- Well-founded rewriting systems, descriptive properties of languages accepted by Buchi automata and Linear Time Logic.
F. Set theory and philosophy. The general objectives are the applications of logical and mathematical methods in philosophy, as well as the analysis of philosophical questions arising in the exact study of infinity. The search for new axioms beyond ZFC that are strong enough to resolve outstanding questions has been remarkably successful over the last 40 years producing, among other things, complete solutions to the classically unanswerable problems of second order number theory. With regard to third order number theory (which is where the Continuum Hypothesis belongs), a breakthrough was achieved only recently. What these results indicate is that the difficulties are not exclusively mathematical and any solution of the continuum problem will have to be accompanied by an analysis of what it means to be a solution. Here philosophical reflections on the nature of infinite mathematical objects and the concept of truth in mathematics are likely to play a role. Our aim is to lay the groundwork for the above analysis and to map out possible directions it may take. Clearly this calls for a transdisciplinary approach. To this end, philosophers need to work together with set theorists and general mathematicians.