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

interpretation of b + vsde + ssde in z 147We now give an interpretation of B + VSDE + SSDE in Z. As remarked in Section6.6, the construction there requires all of the V (ω × n),n) be an irreflexive transitive relation,where E ⊆ V (ω + n),n ∗∗ ) is an irreflexive transitive relationsuch that:i. E ∗ ⊆ V (ω + n + 6).ii. E ⊆ E ∗ ,>⊆> ∗∗ .iii. x ∈ E →(x > ∗∗ y ↔ x>y).iv. Any finite subset of E ∗ has a strict upper bound in (E ∗ , > ∗∗ ).proof Let (E,>) ∈ V (ω + n),n ∗∗ β as follows:Case 1. α, β ∈ E. Then α> ∗∗ β ↔ R(β, α).Case 2. α/∈ E, β ∈ E. Letα = (E,(x 1 ,...,x k )). Then α> ∗∗ β ↔ (∃i)(β = x i )).Case 3. α, β/∈ E. Letα = (E, (x 1 ,...,x k )), β = (E, (y 1 ,...,y n )). Then α> ∗∗ β ↔(x 1 ,...,x k ) is a (not necessarily consecutive) subsequence of (y 1 ,...,y n ).Case 4. α ∈ E, β/∈ E. Then ¬α > ∗∗ β.We represent a finite sequence (x 1 ,...,x k ) from E by a function with domaink. Thus, the elements of the finite sequence are of the form {{i}, {i, x}},where x ∈ A, and so lie in V (ω + n + 2). Hence, the finite sequencesfrom E all lie in V (ω + n + 3). Therefore, each (E,(x 1 ,...,x n )) ={{E},{E,(x 1 ,...,x n )}} lies in V (ω + n + 5). Hence, E ∗ ∈ V (ω + n + 6).We are now ready to define pairs (E n , > n ), for all integers n 0 =⊘. Suppose (E n , > n ) has been defined. We now define (E n+1 , > n+1 ).We define E n+1 = E ∗ n ∪ {(n, 0,A): A is (E ∗ n , > ∗∗ n ) transitive}. We define x> n+1y ↔ x> ∗∗n y ∨ (∃A)(x = (n, 0,A) ∧ y ∈ A).We use (n, 0,A) instead of (n, A) because (n, 0,A) is an ordered triple, and so wecannot duplicate ordered pairs in the construction.We can now create E ω =∪ n ω =∪ n n , as proper classes in Z.We define ≫ by x ≫ y ↔ (∃n)(y ∈ En ∗ ∧ (∀w ∈ E n+1)(x > ω w)).We claim that Z proves that (E ω ,> ω , ≫) forms a model of B + VSDE + SSDE.More precisely, for each axiom of B + VSDE + SSDE, Z proves that the axiom holdsin (E ω ,> ω , ≫).The verification of Basic ′ is straightforward, except for the last assertion.Lemma 7.2 (E ω ,> ω , ≫) satisfies Basic ′ .proof This follows the proof of Lemma 6.3, except for the last axiomof Basic ′ . Let x ≫ y,z. Let y ∈ En ∗, (∀w ∈ E n+1)(x > ω w). Let z ∈ E m ∗,