Souslin Trees and Degrees of Constructibility
by
Francois Dorais
Dartmouth College

Given a set of ordinals A we can relativize Goedel's constructible universe L to form the minimal universe L[A] which contains A as a set. It is natural to ask whether another set of ordinals B is constructible from A, i.e. whether or not B ∈ L[A]. This notion gives us a pre-ordering of the sets of ordinals, and the equivalence classes in this pre-ordering are called degrees of constructibility.

The structure of degrees of constructibility is trivial if V = L, but it can be very rich in forcing extensions of L or in the presence of large cardinals. There are many known restrictions on what this structure can be, and many known forcing constructions which produce rich degree structure. However, there is still a gap between the two.

We will give a combinatorial characterization of the degrees of constructibility in forcing extensions of L via Souslin trees. Then we will use this characterization to construct forcing models in which the degrees of constructibility are isomorphic to constructible lattices which satisfy certain algebraic conditions.

Date received: March 16, 2006