Sacks Forcing and the Shrink Wrapping Property

We consider a property stronger than the Sacks property which holds between the ground model and the Sacks forcing extension. 1. The Shrink Wrapping Property Suppose that V is a Sacks forcing extension of a model M . Then the Sacks property holds between V and M . That is, for each x ∈ ω, there exists a tree T ⊆ ω in M such that x ∈ [T ] and each level of T is finite. The following is a stronger property that we might want to hold between V and M : for every sequence X = 〈xn ∈ ω : n < ω〉 there exists a sequence of trees 〈Tn ⊆ ω : n < ω〉 such that 1) (∀n ∈ ω)xn ∈ [Tn]; 2) (∀n1, n2 ∈ ω) one of the following holds: a) xn1 = xn2 ; b) [Tn1 ] ∩ [Tn2 ] = ∅. Unfortunately, if the sequence X is such that 〈(n1, n2) : xn1 = xn2〉 6∈M, then there can be no such sequence of trees in M . Thus, we need a weaker notion: a shrink wrapper. In this next definition, we fix a canonical bijection η : ω → [ω] so that for each ñ ∈ ω, we may talk about the ñ-th pair η(ñ) ∈ [ω]. The idea is that for each {n1, n2} = η(ñ) ∈ [ω], the functions Fñ,n1 and Fñ,n2 , together with the finite sets I(n1) and I(n2), separate xn1 and xn2 as much as possible. For n ∈ η(ñ), the function Fñ,n : 2→ P(ω) is shrink-wrapping 2 possibilities for the value of xn. We need to make sure that what contains one possibility for xn1 is sufficiently disjoint from what contains another possibility for xn2 , even if it is not possible that simultaneously both xn1 and xn2 are in the respective containers. The main idea of shrink-wrapping is that for a fixed ñ, if {n1, n2} is the ñ-th pair, the trees Fñ,n1(s) for s ∈ 2 and Fñ,n2(s) for s ∈ 2 separate xn1 from xn2 as much as possible. If xn1 = xn2 , they 1