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

some implications 141two disjuncts imply that a is not > ex E.Ifa < z, then by (4) a ∈ E, and so nota > ex E.We claim that(7) P (w ′ ,x ′ ,y ′ ,z ′ ,u ′ ,v ′ ) → u ′ ,v ′ are incomparable.Suppose P (w ′ ,x ′ ,y ′ ,z ′ ,u ′ ,v ′ ). Suppose u ′ ≤ v ′ .By(5),x ′ ≤ y ′ ∨ x ′ ≤ z ′ .Thiscontradicts (5).Suppose v ′ ≤ u ′ . Then y ′ ,z ′ ≤ x ′ . This contradicts (5).We claim that(8) P (w ′ ,x ′ ,y ′ ,z ′ ,u ′ ,v ′ ) ∧ P (w ′ ,x ′′ ,y ′′ ,z ′′ ,u ′′ ,v ′′ ) → (x ′ ,y ′ ,z ′ ,u ′ ,v ′ ) = (x ′′ ,y ′′ ,z ′′ ,u ′′ ,v ′′ ).Suppose P (w ′ ,x ′ ,y ′ ,z ′ ,u ′ ,v ′ ) ∧ P (w ′ ,x ′′ ,y ′′ ,z ′′ ,u ′′ ,v ′′ ). Then w ′ > ex {u ′ ,v ′ },w ′ > ex {u ′′ ,v ′′ }. By (7), u ′ ≤ u ′′ ∨ u ′ ≤ v ′′ ,v ′ ≤ u ′′ ∨ v ′ ≤ v ′′ ,u ′′ ≤ u ′ ∨ u ′′ ≤v ′ ,v ′′ ≤ u ′ ∨ v ′′ ≤ v ′ .Case 1. u ′ ≤ u ′′ .By(7),u ′′ ≤ u ′ ,u ′ = u ′′ .Case 2. u ′ ≤ v ′′ .By(7),v ′′ ≤ u ′ ,u ′ = v ′′ .By symmetry, v ′ = v ′′ ∨ v ′ = u ′′ . Hence, {u ′ ,v ′ }={u ′′ ,v ′′ }.Suppose u ′ = v ′′ . Since y ′′ ,z ′′ are incomparable, we have x ′ ≤ y ′′ ∨ x ′ ≤ z ′′ ,y ′′ ≤x ′ ,z ′′ ≤ x ′ . Hence, x ′ = y ′′ ≤ x ′ ∨ x ′ ≤ z ′′ ≤ x ′ . Therefore, x ′ = y ′′ ∨ x ′ = z ′′ .Thiscontradicts the incomparability of y ′′ ,z ′′ . Hence, u ′ ≠ v ′′ . Therefore, u ′ = u ′′ ∧ v ′ =v ′′ . From the first conjunct, x ′ = x ′′ .By the incomparability of y ′ ,z ′ , and of y ′′ ,z ′′ , we have {y ′ ,z ′ }={y ′′ ,z ′′ }.Suppose y ′ = z ′′ . Then x ′ >z ′′ , and so x ′′ >z ′′ , which is impossible. Hence, y ′ =y ′′ ∧ z ′ = z ′′ .We now define(9) R(w ′ ,b,c) ↔ (∃x ′ ,y ′ ,z ′ ,u ′ ,v ′ )(P (w ′ ,x ′ ,y ′ ,z ′ ,u ′ ,v ′ ) ∧ c = y ′ ∧ (b