Inner Model Theory in Outer Models

The research of the proposed project is within axiomatic set theory. This theory is usually seen as a basis for all of mathematics, since every mathematical concept can be expressed structurally in terms of sets. Much of what mathematicians do concerns infinite sets or collections and it is this notion of 'infinite' that set theorists try to elucidate. Although the world is of finite size, the theoretical effects of the infinity of counting numbers 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.

In set theory much fundamental work was done by Kurt Goedel in showing that certain axioms known as the Axiom of Choice and the property known as Cantor's Continuum Hypothesis (CH - that every set of numbers on the number line is either countable or of the same size as the whole line) were consistent with the universally accepted axiom set. He did this by developing a structure or 'inner model' of those axioms with those desired extra properties. This process of inner model building has come to be seen as fundamental to our understanding of the universe of all sets of mathematical discourse (known as 'V').
Goedel's structure, called 'L', is now widely generalised and strengthened to incorporate more and more potential properties, or stronger axioms, that may hold in V. This program of inner model analysis and building was initiated by Ronald Jensen in the 1970's, who discovered fundamental properties of 'L' (called its 'fine structure').

However Paul Cohen in 1962 showed by a radically new method called 'forcing' that this could not be the whole story: one could build syntactic or 'virtual' models of the axioms in which properties such as the CH failed: such properties we call independent of the axioms.

The research being undertaken here is very novel in that it tries to ascertain to what degree the fine structure of inner models can hold in certain of these 'virtual' (which we call 'outer' in the project) models. These outer models are often built assuming strong axioms hold in V, and theorists using the forcing techniques try and preserve these axioms when building them. But is it possible to have such strong axioms with at the same time fine structure of an inner model? Or are they incompatible? This is broadly the question that this project wishes to investigate.

Why should we be concerned about this? From the viewpoint of set theorists this is important as the program building inner models has run into difficulties, and model building is (perhaps temporarily) halted. We might ask: are there then mathematical reasons for this? This project can help elucidate fundamental incompatibilities (if any) between fine structure and strong axioms. But 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. The area has been recognised with a large European Research Grant called INFTY. However,in the UK advanced set theory is somewhat underrepresented, and is concentrated in Bristol and at UEA. This project will thus enhance the UK's standing and expertise in set theory.

This project concerns fundamental research in the logical foundations of the one activity that underpins all scientific endeavour (whether a physical, medical or a social science) namely mathematics. It is thus, in short, a rather pure project dealing with the intellectual knowledge concerning the nature of mathematics.

We thus see it as contributing to the "knowledge economy" of that part of the academic community who study such fields as set theory. In general the impact of such projects is not measured in the next immediate few months or years, but rather decades. It increases our understanding of some of the most basic issues around the foundations of mathematics.

The beneficiaries of this research will hence be those, mainly in the academic community, who think or research on these issues: mostly set theorists and those in neighbouring fields, where some of the issues we look at have an impact: combinatorics and to an extent philosophers of mathematics, who have a technical knowledge and interest in foundational issues.

To maximise this impact we shall hold a Concluding Workshop on this research, as well as disseminate our ideas through the usual means of publication in first rate academic journals and at conferences to which we shall speak on this work.

The Research Environment and Training:

The project will enhance the UK's knowledge in the field under study, which concentrates mainly (but not entirely) on a new development in "inner model theory" emerging from the Kurt Goedel Research Institute for Mathematical Logic in Vienna, from which we wish to study emergent new techniques, and hence enhance the knowledge base of the UK in this area.
The project will moreover act as a decisive step in the career of the younger Researcher we shall need to employ on this project, as well as enhancing the Post-Graduate research environment for current PhD students at our department, and for the subject within the UK. It is to be hoped that through seminars and active collaboration and interaction the RA will have a positive impact on the training of those PhD students. In turn there will be knowledge transfer to the RA from working closely with the PI and, as we expect, in particular contact with UEA on forcing techniques.