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

140 concept calculus: much better than6.5 Some ImplicationsWe establish a number of implications, some of which are needed for Section 6.9.Theorem 5.1SDE → DE.The following are provable in B. SSDE → SDE → DE. VSDE →proof Assume SSDE. Let x ≫ y > S. Then x ≫ S, and apply SSDE.Assume SDE. Let x > S. Let y ≫ x > S. By SDE, let z > ex S, ¬z < y. Then¬z < x. The last claim is immediate.Recall that MBT = B + DE + SUI.We say that a is >-equivalent to b if and only if (∀x)(a > x ↔ b > x).We say that a is >-included in b if and only if (∀x)(a > x → b > x).We say that a, b are incomparable if and only if a ≠ b ∧¬a > b ∧¬b > a.Theorem 5.2MBT proves B + SDE + UI.proof It is immediate that MBT proves B + UI. We have only to show thatMBT proves SDE. The diversity in SDE follows from DE. Let(1) x ≫ y > E.Without loss of generality, assume that(2) E is > transitive.Assume that(3) nothing < x is > ex E.We derive a contradiction.By DE, let(4) z > ex E, ¬x > z, u > ex {x}, v> ex {y, z}, w> ex {u, v}.Define(5) P (w ′ ,x ′ ,y ′ ,z ′ ,u ′ ,v ′ ) ↔ x ′ >y ′ ∧ y ′ ,z ′ are incomparable ∧x ′ ,z ′ are incomparable∧u ′ > ex {x ′ }∧v ′ > ex {y ′ ,z ′ }∧w ′ > ex {u ′ ,v ′ }∧z ′ is >-included iny ′ ∧ (∀a < b -equivalent to z ′ ).We claim that(6) P (w, x, y, z, u, v).Suppose y, z are comparable. By (1), (4), x > y ∧¬x > z. Hence z > y.By(4),y ∈ E.By (1), y > E. Hence y > y, which is a contradiction. Hence y, z are incomparable.Suppose x, z are comparable. Since y, z are incomparable and x > y, wehavex > z. This contradicts (4).By (1), (4), z is >-included in y. Hence, we have only to verify that (∀a < b -equivalent to z). By (4), we need only verify that (∀a < b ex E).Suppose a < b ex E.By(4),a < b ≤ u ∨ a < b ≤ v. Hence, a