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

138 concept calculus: much better thanStrong Unlimited Improvement (SUI). Let x and a ternary relation be given.There are arbitrarily good y such that x, y are each related, by the given ternary relation,to the same pairs of things that x is much better than.In SUI, binary relations are given by formulas in L(>, =) with no side parameters.We place the greatest emphasis on the system MBT (much better than) = B +DE + SUI. We prove that MBT is mutually interpretable with ZF.We isolate the following three strengthenings of Diverse Exactness:Strong Diverse Exactness (SDE). Let x be much better than something betterthan a given range of things. Then x is better than some, but not all, things exactlybetter than the given range of things.Very Strong Diverse Exactness (VSDE). Let x be much better than somethingbetter than a given range of things. Then x is much better than some, but not all, thingsexactly better than the given range of things.Super Strong Diverse Exactness (SSDE). Let x be much better than something,and a given range of things. Then x is better than some, but not all, things exactly betterthan the given range of things.In these three, ranges of things are given by formulas in L(>, ≫, =), with sideparameters allowed.We derive SDE from B + VSDE and from B + SSDE.We also derive SDE and UI from MBT.We show that B + SDE is mutually interpretable with Z. Here Z is Zermelo settheory, a fragment of ZF of considerable strength. This result holds for B + VSDE +SSDE.We show that B + SDE + UI is mutually interpretable with ZF.We also show that all seven of our axioms (axiom schemes), together, is mutuallyinterpretable with ZF(CTo avoid any ambiguities, we now present these axioms formally.Let ϕ be a formula in L(>, ≫, =), where y is not free in ϕ. We definey>ϕiff (∀x)(ϕ → y>x).y ≫ ϕ iff (∀x)(ϕ → y ≫ x).y> ex ϕ iff (∀z)(y >z↔ (∃x)(ϕ ∧ (x = z ∨ x>z))).Note that in the preceding formulas, we think of ϕ = ϕ(x). Thus, the variable x has aspecial status. The other variables, y, z, are allowed to be any distinct variables, otherthan x, that do not appear in ϕ.Basic (B). ¬x>x.x>y∧ y>z→ x>z.x≫ y → x>y.x≫ y ∧ y>z→ x ≫z. x > y ∧ y ≫ z → x ≫ z. (∃z)(z ≫ x ∧ z ≫ y). x≫ y → (∃z)(x ≫ z ∧ z>y).Diverse Exactness (DE). y >ϕ→ (∃z)(¬y > z ∧ z > ex ϕ), where ϕ is a formulain L(>, ≫, =) in which y, z are not free.