Seminar series
Date
Mon, 14 Nov 2005
15:30
15:30
Location
SR2
Speaker
Robin Knight
Organisation
Oxford
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.
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.