**S. Homer and W. Maass**

We consider under the assumption questions concerning the structure
of the lattice of sets together with the sublattice . We show that
two questions which are slightly more complex than the known splitting
properties of this lattice cannot be settled by arguments which relativize.
The two questions which we consider are whether every infinite set
contains an infinite subset and whether there exists an -simple set.
We construct several oracles, all of which make , and which in
addition make the above-mentioned statements either true or false. In
particular we give a positive answer to the question, raised by Bennett and
Gill 1981), whether an oracle exists making and such that
every infinite set in has an infinite subset in . The
constructions of the oracles are finite injury priority arguments.