Graphs on Generalised Baire Spaces

The research of the proposed project is within axiomatic set theory. This theory is usually seen as a foundation for all of mathematics, since every mathematical concept can be expressed structurally in terms of infinite sets. Although the world is of finite size, the theoretical effects of the infinity of our counting numbers, ``N'', is felt through, eg, modelling of computation by programs and numbers as discovered by Turing: although computers are finite, theorizing about their capabilities is best done in an infinite context. In similar ways we model the finite world by using 'infinite structures' and theories.
G. Cantor, the originator of modern set theory, tried to solve knotty problems about subsets of the real number line by establishing results first for simply described sets, then building up for more complicated ones, etc. This founded the concept of 'descriptive set theory'.
In the `classical period' of the 1910's and 20's the Russian (Suslin, Luzin) and French (Borel, Lebesgue) schools of analysts worked intensively on establishing results up to the level they could describe: 'Borel' or 'analytic' sets. For example, for these sets in the plane an idea of "area'' can be developed even if these are not regions enclosed by a simple curve. However matters were stuck at this level. Lebesgue had defined a hierarchy of "projective sets'' beyond the analytic, but despaired of discovering whether they could be 'measurable' in this sense. Modern set theory has discovered why the classical analysts were stuck: axioms, or postulates, beyond the standardly used ones of Zermelo-Fraenkel (developed in the 1920's "ZF'') were needed. Either stronger "axioms of infinity" (also called "large cardinals'') were needed to be assumed in the universe of sets to get these projective sets to behave properly. One surprising but significant development was the use of infinitely long two person perfect information games. Assuming such games had winning strategies played a role. Players alternated integer moves, and the games had length the same type as N. These are technically known as "games on Baire space''.

Our project is to refocus some of these ideas on a current new area of interest that has sprung up: "Generalised Baire spaces'': instead of sequences of type N that can be construed as a decimal expansion of a real number, we look at yet longer sequences the type of one of Cantor's large uncountable cardinal numbers, that is yet greater than the size of the natural numbers. The associated conceptual games are also longer in this sense, and may, or may not, be susceptible to the same kinds of analysis as the earlier ones. We do not yet know. The original Baire space is often identified with the irrational numbers (the countably many rationals left over not counting towards notions of area, measure etc.) We can thus think of the Generalised versions as generalising the real number line in this particular direction.

Why should we be concerned about this? The implication of studying such stronger axioms are much wider: for the general mathematical analysts strong axioms affect how they view the real number line, and this is only now starting to be appreciated. Several areas of pure mathematics can be said to be directly affected by set theoretic axiomatics. In the wider perspective an understanding of the nature of 'infinity' and 'set' is of interest both philosophically and for the general human endeavour. We thus think of the beneficiaries of this research as principally set theorists, but more widely,
mathematical logicians and philosophers of mathematics who are interested in these questions.

Set Theory is very active internationally, with significant research groups in, eg, USA, Israel, Austria, France, Germany. However, in the UK advanced set theory is somewhat underrepresented, and is concentrated in Bristol, UEA and at Leeds. This project will thus enhance the UK's standing and expertise in set theory.