| Résume | There are many natural set-theoretic structures which satisfy every axiom of ZFC except for Power Set, for example the collection of hereditarily countable sets. Therefore, it is worth investigating what this theory is. If one simply deletes the Power Set axiom but uses the standard formulation of the other axioms (in particular with only the Replacement Scheme) then this theory may have unexpected consequences, for example it is possible that omega_1 exists but is singular. On the other hand, most natural structures satisfy the stronger scheme of Collection, which prevents many of these undesired possibilities from occurring. Therefore, the standard definition of this theory is "ZFC without Power Set but with the Collection Scheme".
In this talk we are going to investigate some limits of this stronger theory by considering the notion of a big proper class, which is a proper class that surjects onto every non-zero ordinal. We shall see that, even with Collection, there are models of ZFC without Power Set in which the reals form a proper class that is not big. However, if one additionally assumes the schemes of Dependent Choices for arbitrary lengths, then every proper class is indeed big. Building on work of Zarach, we will provide a general framework for separating Dependent Choice schemes of various lengths. Finally, using a similar idea, we will produce a new model of ZFC without Power Set but with Collection in which there are unboundedly many cardinals, but the Reflection Principle (and therefore the scheme of Dependent Choice of length omega) fails. This is joint work with Victoria Gitman. |
