Hinges and Automorphisms of the Degrees of Non‐Constructibility

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 ^ c a then either b = c 0 or b = c a.