Constructible lattices of c-degrees

Extract

This paper is devoted to the proof of the following theorem.

Theorem. Let M be a countable standard transitive model of ZF + V = L, and let ℒ Є M be a wellfounded lattice in M, with top and bottom. Let ∣ℒ∣^M = λ, and suppose κ ≥ λ is a regular cardinal in M. Then there is a generic extension N of M such that

(i) N and M have the same cardinals, and ^κN ⊂ M;

(ii) the c-degrees of sets of ordinals of N form a pattern isomorphic to ℒ;

(iii) if A ⊂ On and A Є N, there is B Є P(κ⁺)^N such that L(A) = L(B).

The proof proceeds by forcing with Souslin trees, and relies heavily on techniques developed by Jech. In [5] he uses these techniques to construct simple Boolean algebras in L, and in [6] he uses them to construct a model of set theory whose c-degrees have orderlype 1 + ω*.

The proof also draws on ideas of Adamovicz. In [1]–[3] she obtains consistency results concerning the possible patterns of c-degrees of sets of ordinals using perfect set forcing and symmetric models. These methods have the advantage of yielding real degrees, but involve greater combinatorial complexity, in particular the use of ‘sequential representations’ of lattices.

The advantage of the approach using Souslin trees is twofold: first, we can make use of ready-made combinatorial principles which hold in L, and secondly, the notion of genericity over a Souslin tree is particularly simple.