This led to the study of systems of finite type which still allow to formalize substantial parts of analysis, but which also guarantee the extractability of primitive recursive, elementary recursive and even polynomial data. The extraction process is based on a monotone functional interpretation, a new method to exhibit the computational content of fragments of arithmetical comprehension and the use of non-standard axioms. We also show the necessity to use functionals of arbitrary finite types for modular and at the same time faithful proof interpretations. Last time, we went as far as it's reasonably possible to go by starting from the bottom and going up. This time, I'll talk about some of the other ways of trying to define the boundary.
Measurable cardinals are interesting for a variety of reasons.
One reason is that it took twenty-five years to find out that they were actually large. The existence of a measurable cardinal also implies some slightly startling things about sets of integers in the constructible universe. In these talks, I'll take you on a trip round the various boundary posts, which are increasingly large cardinals with increasingly silly names. In the first part of the trip, we start from home, and head towards what looks like the farthest point. The boundary posts we meet on this part are the inaccessible, hyperinaccessible, Mahlo and hyper-Mahlo cardinals.