Souslin Trees and Degrees of Constructibility

This thesis investigates possible initial segments of the degrees of constructibility. Specifically, we completely characterize the structure of degrees in generic extensions of the constructible universe L via forcing with Souslin trees. Then we use this characterization to realize any constructible dual algebraic lattice as a possible initial segment of the degrees of constructibility. In a seminal paper [20], Gerald E. Sacks introduced perfect set forcing (also known as Sacks forcing) to show that the two-element lattice can be realized as an initial segment of the degrees of constructibility; i.e., if S is a perfect set generic over L, then L[S] has precisely two degrees of constructibility. Refining Sacks’s method, Marcia J. Groszek and Richard A. Shore have shown that for every countable complete algebraic lattice L ∈ L, there is a notion of forcing such that the degrees in the generic extension form a lattice dual isomorphic to L. Using a completely different method — namely forcing with Souslin trees — we extend this result by removing the size restriction. Theorem. (1.1.7) Assume V = L. Let κ be an infinite regular cardinal and let L be a complete algebraic lattice with at most κ compact elements. There is a Souslin tree T of height κ such that if G is a generic branch through T , then the degrees of constructibility in L[G] form a lattice dual isomorphic to L. In view of this result, a natural question is: if the c-degrees in L[A] form a complete lattice, must the lattice be algebraic? We show that this is not the case.