Hinges and Automorphisms of the Degrees of Non-Constructibility
Paddy Farrington · Journal of the London Mathematical Society · 1983
The main result of this paper is to show that, under weak cardinal assumptions, there is no non-trivial automorphism of the degrees of non-constructibility. To achieve this we introduce the notion of a hinge. The degrees of non-constructibility, or c-degrees, are the factor classes of the reals (in some cases we shall consider larger sets of ordinals) under the following equivalence relation: a =c b if and only if a e L(b) and b e L(a). Given a real number a we define its degree a to be the set {b ^ a>: a =c b). The set of c-degrees, which we call C, is then ordered by the following relation: a ^c b if and only if a e L(b). For any degrees a, b we define a v b as the degree of the pair (a, b). A c-degree a is said to be minimal if, for any degree b, whenever b ^ca then either b =c 0 or b =c a. 1. Hinges DEFINITION 1.1. A degree a is a hinge if ^ a)(3c)(b =ca v c). A degree a is a strong hinge if c can be chosen to be minimal. DEFINITION 1.2. A degree a is a collapsing degree if wj " \\ such that, in the forcing extension, V = L(a) where a is a minimal real degree. There are coj dense open subsets of P in L. If d is a collapsing degree, let d enumerate these subsets as a sequence >. Then define a function /:2 Pas follows: f(0) = 2 , ba = f] {/(s): s c = a} is L-generic over P, and hence is minimal. Finally, for any degree a c$s d, bfl v d =c a.