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

etter than, much better than 139Unlimited Improvement (UI). x ≫ y ∧ ϕ → (∃x)(x >w∧ w)), where ϕ is aformula of L(>, =) with at most the free variables x, y.Strong Unlimited Improvement (SUI). (∃y)(y >z∧ (∀v, w)(x ≫ v, w →(ϕ ↔ ϕ[x/y]))), where ϕ is a formula of L(>, =) with at most the free variablesx,v,w, in which y does not appear.Strong Diverse Exactness (SDE). y ≫ z>ϕ→ (∃z)(y>z> ex ϕ) ∧ (∃z)(¬y>z ∧ z> ex ϕ), where ϕ is a formula in L(>, ≫, =) in which y, z are not free.Very Strong Diverse Exactness (VSDE). y ≫ z>ϕ→ (∃z)(y ≫ z> exϕ) ∧ (∃z)(¬y ≫ z ∧ z> ex ϕ), where ϕ is a formula in L(>, ≫, =) in which y, zare not free.Super Strong Diverse Exactness (SSDE). y ≫ ϕ ∧ (∃x)(y ≫ x) → (∃z)(y >z> ex ϕ) ∧ (∃z)(¬y >z∧ z> ex ϕ), where ϕ is a formula in L(>, ≫, =) in which y, zare not free.We defineMBT = B + DE + SUI.We take Z to be pairing, extensionality, union, power set, separation, and infinity. Itis customary to take the axiom of infinity to be in either of two forms.The first is the same weak form that is most commonly used in ZF. It asserts theexistence of a set containing the element ⊘ and closed under the operation that sendsx to x ∪ {x}. It is well known that this form of Z does not suffice to prove the existenceof V(ω).The second form is the stronger form, based on the operation that sends x, y to x ∪{y}. This form of Z does suffice to prove the existence of V(ω).What happens to Russell’s paradox in this context? In sets, we start withThere is a set whose elements are exactly the setswith a given property.and obtain a contradiction that Frege missed and Russell saw. The correspondingprinciple here isThere is something which is better than, exactly,the things with a given property andthose things they are better than.This immediately leads to a contradiction. Even the much weakerThere is something which is better than the thingswith a given property.gives an immediate contradiction, because there cannot be anything that is better thanall things, by irreflexivity. In other words, nothing can be better than itself.Thus, Russell’s Paradox now becomes entirely transparent and never would havetrapped anyone: it disappears as a paradox. Clearly, there is no residual feeling ofmystery here, as there is in the context of sets and properties.