On coding uncountable sets by reals

The following is our main result. Theorem 1.1 Suppose that A ⊆ ω1 and = [A]. Then there exists a cardinal preserving generic extension [x] of the ground universe by a generic real x such that (i) A ∈ [x] – this implies [x] = [x], and A ∈ Δ 1 (x) in [x], (ii) x is a minimal real over , that is, x ∈ , and if a set Y belongs to [x], then x ∈ [Y ] or Y ∈ , 1) (iii) there is a club C ∈ [x], C ⊆ ω1, that reshapes A, i.e., if α ∈ C, then α < ω [A∩α] 1 . We may compress the properties (i) and (ii) of a real x in the theorem by saying that x minimally codes the set A ⊆ ω1. Jensen and Solovay [12] proposed a method of coding of uncountable sets by reals by means of almost disjoint forcing. In the context of Theorem 1.1, this coding method consists of two parts. The first part is the reshaping of A by means of a generic club (closed and unbounded set) C ⊆ ω1 with the properties that 1) C does not add new reals to [A], and 2) if ξ ∈ C is a limit ordinal, then ξ < ω [A∩ξ] 1 (see Theorem 14.1 below). After this is done, a type of almost-disjoint ccc forcing is employed to produce a generic real x over [A][C] such that A and C belong to [x], that is, x codes those two sets. These methods were expanded to a greatly more complicated technique of coding the universe [5]. The almost-disjoint forcing technique does not provide minimal reals. The first result on minimal coding was published in [13]: a generic minimal upper bound of the constructibility degrees of any model satisfying CH, with the minimality understood only in the sense of reals (weak minimality in discussions below). The coding technique in [13] involves a subforcing of the Sacks forcing close to a forcing notion introduced in [11]. 2)