Some Weakenings of CH
by
Kenneth Kunen
University of Wisconsin

This talk describes recent work of Kunen and Juhász, but will also survey earlier work by many other people.

There are many consequences of CH which are independent of ZFC, but are still true in Cohen extensions of models of CH. A number of more general principles are known which embody these consequences. Roughly, these principles fall into two classes. In one class are elementary submodel axioms, saying that there are elementary submodels M of the universe such that M has size \aleph₁ and the reals of M ``capture'' in some way all reals. In the other are homogeneity axioms, saying that given an omega<sub>2</sub> sequence of reals, there are \omega<sub>2</sub> of them which ``look alike''.

Both classes of axioms are trivially true under CH.

We define a new principle, SEP, which is true in all Cohen extensions of models of CH, and explore the relationship between SEP and other such principles.

Date received: February 29, 2000