Past Analytic Topology in Mathematics and Computer Science

 

We consider as a starting point a problem raised by Kunen and Tall as to whether

the real continuum can have non-homeomorphic versions in different submodels of

the universe of all sets. Its resolution depends on modest large cardinals.

In general Junqueira and Tall have made a study of such "substructure spaces"

where the topology of a subspace can be different from the usual relative

topology.

We will survey the topological interpretation of modal languages, with some modern features, such as the appropriate bisimulations and model comparison games. Then we move to an epistemic version of this, showing how it provides a finer set of epistemic distinctions for group behaviour, including different notions of common knowledge. We explain the background for this in an epistemic MU-calculus. Finally, if we can pull this off within the time limit, we will discuss how topological models also show up in current dynamic-epistemic systems of belief revision.

 

 

With a theory in a logical language is associated a {\it type category}, which is a collection of topological spaces with appropriate functions between them. If the language is countable and first-order, then the spaces are compact and metrisable. If the language is a countable fragment of $L_{\omega_1,\omega}$, and so admits some formulae of infinite length, then the spaces will be Polish, but not necessarily compact.

We describe a machine for turning theories in the more expressive $L_{\omega_1,\omega}$ into first order, by using a topological compactification. We cannot hope to achieve an exact translation; what we do instead is create a new theory whose models are the models of the old theory, together with countably many extra models which are generated by the extra points in the compactification, and are very easy to describe.

We will mention one or two applications of these ideas.