Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

interpretation of mbt in zf 145structure (D λ ,> λ , ≫ S ). D λ ,> λ have already been defined. We define x ≫ S y,if and only ifRemarkx,y ∈ D λ ∧ (∃α, β ∈ S)(α λ w)).We could have defined x ≫ S y, if and only ifx,y ∈ D λ ∧ (∃α ∈ S)(α λ w)),but then it can be shown that under this definition, Strong Diverse Exactnesswould fail in M[S].Lemma 6.3M[S] satisfies Basic ′ + VSDE.proof Because (D λ ,> λ ) is irreflexive and transitive, we have the first twostatements in Basic. The third is obvious.Suppose x ≫ S y,y > λ z. Let α λ w).Then z ∈ D α , and so x ≫ S z.Suppose x> λ y,y ≫ S z. Let α λ w).Then (∀w ∈ D β )(x > λ w), and so x ≫ S z.Let y,z ∈ D λ . Let y,z ∈ D α . Let βγ.For SSDE, we need a further condition on S. This is because SSDE will fail in M[S]if some element of S is a limit of elements of S.For SSDE, we need a further condition on S. This is because SSDE will fail in M[S]if some element of S is a limit of elements of S.Lemma 6.4Suppose S is of order type ω. Then M[S] satisfies SSDE.proof Let S be as given. Let y ≫ S ϕ, and y ≫ S u for some u. Let B ={x :ϕ(x) holds in M[S]}. First assume B =⊘. Let α λ w). Then y ≫ S (1, ⊘).Now assume B ≠⊘. For each x ∈ B, let α x λ w). Then the β x are bounded below λ, and so theβ x have a max, β. Also, the α x are bounded below λ, and also have a max, α.Obviously α λ w). Clearly (α, B) is an exact upper boundfor B. Since (α, B) ∈ D β , clearly y > λ (α, B).Let y ∈ D γ ,γ λ (β, z).We have given an interpretation of B ′ + VSDE + SSDE in ZF. Actually, we onlyneed V(ω 2 ) for this construction, taking S ={ω,ω × 2,ω× 3,...},λ= ω 2 . Thus, wehave not provided an interpretation of even B + SDE in Z, or Zermelo set theory. In