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

CHAPTER 4The Realm of the InfiniteW. Hugh Woodin4.1 IntroductionThe twentieth century witnessed the development and refinement of the mathematicalnotion of infinity. Here, of course, I am referring primarily to the development of settheory, which is that area of modern mathematics devoted to the study of infinity. Thisdevelopment raises an obvious question: is there a nonphysical realm of infinity?As is customary in modern set theory, V denotes the universe of sets. The purposeof this notation is to facilitate the (mathematical) discussion of set theory – it does notpresuppose any meaning to the concept of the universe of sets.The basic properties of V are specified by the ZFC axioms. These axioms allow oneto infer the existence of a rich collection of sets, a collection that is complex enoughto support all of modern mathematics (and this, according to some, is the only point ofthe conception of the universe of sets).I shall assume familiarity with elementary aspects of set theory. The ordinals calibrateV through the definition of the cumulative hierarchy of sets (Zermelo 1930). Therelevant definition is given as follows:Definition 1. Define for each ordinal α a set V α by induction on α.(1) V 0 =∅.(2) V α+1 = P(V α ) = {X | X ⊆ V α }.(3) If β is a limit ordinal, then V α =∪{V β | β