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

146 concept calculus: much better thanSection 6.6 we will modify M[S] so that it does provide an interpretation of B ′ +VSDE + SSDE in Z.We now come to Strong Unlimited Improvement. Here we need further conditionson S, even for Unlimited Improvement.Lemma 6.5 For all k ≥ 1 there exists r ≥ 1 such that the following holds.Suppose S has order type ω, where for all α from S, V(α) is an elementarysubmodel of V(λ) for formulas with at most r quantifiers. Then M[S] is a modelof k-SUI, which is SUI in which the formula ϕ has at most k quantifiers.proof By Lemmas 6.3 and 6.4, it suffices to verify that k-SUI holds inM[S]. Let k, r, S be as above. Unless stated otherwise, ϕ is always evaluated in(D λ ,> λ ). Let ϕ(u, y, z) define the ternary relation for k-SUI, for “u is related toy, z”. Choose r to be 100(k + 1). We will evaluate ϕ in M[S].Let x be given. We can assume that (∃v)(x ≫ S v). Let β ∈ S be greatestsuch that (∀w ∈ D β )(x > λ w). Then β is not the least element of S. Let α bethe preceding element of S. Let E ={: x ≫ S y,z ∧ ϕ(x,y,z)}. ThenE ={ ∈ V (α) :ϕ(x,y,z)}. Then E ∈ V (β). We have(1) (∃x)((∀w ∈ D β )(x > λ w) ∧ (∀y,z ∈ E)(ϕ(x,y,z))).It suffices to show that β can be replaced by any ordinal