More on proper forcing

Extract

§1. A counterexample and preservation of “proper + X”.

Theorem. Suppose V satisfies, , and for some A ⊆ ω₁, every B ⊆ ω₁, belongs to L[A].

Then we can define a countable support iterationsuch that the following conditions hold:

a) EachQ_iis proper and ⊩_Pi “Q_i, has power ℵ₁”.

b) Each Q_i is -complete for some simple ℵ₁-completeness system.

c) Forcing with P_α = Lim adds reals.

Proof. We shall define Q_i by induction on i so that conditions a) and b) are satisfied, and C_i, is a Q_i-name of a closed unbounded subset of ω₁. Let : ξ < ω₁› ∈ L[A] be a list of all functions f which are from δ to δ for some δ < ω₁ and let h: ω₁ → ω₁, h ∈ L[A], be defined by h(α) = Min{β: β > α and L_β[A]⊨ “∣α∣ = ℵ₀”}.

Suppose we have defined Q_j for every j < i; then P_i is defined, is proper (as each Q_j, j < i, is proper, and by III 3.2) and has a dense subset of power ℵ (by III 4.1). Let G_i ⊆ P_i be generic so clearly there is B ⊆ ω₁, such that in V[G_i] every subset of ω₁ belongs to L[A, B], The following now follows:

Fact. In V[G_i], every countableN ⥽(H(ℵ₂), ∈, A, B) is isomorphic toL_β[A ∩ δ, B ∩ δ] for some β < h(δ), where δ = δ(N) = ω₁, ∩ N.