February 25
Richard Matthews,
University of Leeds
Big classes and the respected model
In standard (ZFC) set theory, proper classes are not sets because they are too 'big' or, to put it in a formal way, because they surject onto any non-zero ordinal. We shall study this notion of 'bigness' in weaker systems of set theory, in particular those in which the Power Set Axiom fails. We will observe that in many such theories it is possible to have proper classes which are not big.
As part of this, we shall see a failed attempt to find a proper class which is not big in the theory ZF without Power Set but with Collection - which is by taking a certain symmetric submodel of a class forcing. It will turn out that this approach fails because, unlike in the set forcing case, the symmetric submodel of a class forcing need not exhibit many of the nice properties that we would expect. Notably, Collection may fail and, in fact, it is unclear which axioms need necessarily hold.
This will lead to the definition of the 'Respected Model', an alternative approach to defining a submodel of a class forcing in which Choice fails. We will investigate the properties of this new model and compare it to the symmetric version.