Definable Sets of Minimal Degree

A definable E 0 class containing no definable elements

“…There is a tree T ∈ P , T ⊆ R , and a string u ∈ 2 <ω such that lh(u) > lh((stem(T s ))) and f "[T ] ⊆ [u]. 6 There is a string v ∈ T s incomparable with u . Put S = T s v ; then [S] ∩ f "[T ] = ∅.…”

“…6 Recall that [u] = {a ∈ 2 ω : u ⊂ a} is the Baire interval in 2 ω . 7 The code of a continuous f : 2 ω → 2 ω is the family of sets C t = {u ∈ 2 <ω : f "[u] ⊆ [t]} , t ∈ 2 <ω .…”

“…Definition 6.1 (in L) Following Jensen [6,Section 3], define, by induction on ξ < ω 1 , a countable STF U ξ (Sect. 2), as follows.…”

“…While Jensen's forcing notion in [6] guarantees that there is a single generic real in the extension, the forcing notion P we use adds a whole E 0 -class (a countable set) of generic reals!…”

CH and the Sacks property

“…Many of these results deal with the Sacks property: recall that a poset P has the Sacks property if for any real r in the generic extension there is a sequence of finite subsets of ω, {I n } n ∈ V , such that r ∈ n<ω I n and |I n | ≤ 2 n for any n. Notice that when replacing the function f (n) = 2 n by an arbitrary increasing function we get an equivalent formulation. Jensen deduced in [3] the existence of ccc posets with the Sacks property from 3. Extracting a more general statement about ccc forcings, Veličković showed in [10] that it is possible to have a large continuum and the existence of ccc posets with the Sacks property.…”