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

geometry, infinity, and the peano curve 67we can always choose a member from each set in the collection, was added in whatis called the ZFC model of set theory. The sixth axiom of ZF is the existence of aninfinite set, the acceptance of infinity.The axiom of choice AC was shown, by Kurt Gödel and Paul J. Cohen, to beindependent of ZF. Thus, if ZF is consistent, then not only ZFC is consistent, but alsoZF together with the negation of AC is consistent. Most mathematicians accept AC,but some are not happy with it because it also yields results contrary to our intuitioncoming from the real world. A well-known example is the Banach-Tarski paradox:With AC, it is possible to decompose the closed unit ball in 3-space into five piecesand reassemble these pieces, without deforming them, to form two copies of the sameball. As strange as it may seem, there is no contradiction here. The pieces in questionare not measurable and we cannot, for example, speak of their volume. In essence,what would be achieved by stretching is achieved here by allowing a decomposition intotally weird pieces.What then is the status of CH? Gödel first proved that CH is consistent with ZFC,and finally Paul Cohen proved that its negation, ¬CH, is also consistent with ZFC.In other words, the CH, as well as its negation, can be added to ZFC as an axiom.However, the last word on the subject has not been written yet, and very interestingalternatives to ZFC have been put forward, as with Grothendieck’s universes. In naiveterms, a universe U is a set in which all the operations of set theory can be performed,without ever getting out of U. The existence of a universe U cannot be provedfrom ZF (it is equivalent to the existence of strongly inaccessible large cardinals).Universes provide models of ZF in which it makes sense to speak of “the set of allsets.”2.8 Geometry, Infinity, and the Peano CurveIn the process of this revision of analysis, intuitive concepts such as continuity wentthrough a complete review. Pathological examples, showing that continuity could leadto unforeseen possibilities, were found. Famous examples are Weierstrass’ constructionof a function of one variable, continuous but nowhere differentiable, and Peano’sexample of a continuous map ϕ :[0, 1] → [0, 1] × [0, 1] of the unit interval to the unitsquare, with range the whole unit square. These examples are counterintuitive to thenotion of tangent of a curve, and to the very notion of curve. How was it possible tohave a continuous curve completely filling a square? In a letter of May 20, 1893, tohis friend Stieltjes (Correspondance d’Hermite et de Stieltjes 1905. T.2, letter 374),the great mathematician Charles Hermite wrote, referring to certain series that werenot convergent and complimenting Stieltjes on his expansion in continued fractionof a complicated integral: “Mais ces développements, si élegants, sont frappés demalédiction; leurs dérivees d’ordre 2m + 1 et 2m + 2 sont des séries qui n’ont aucunsens. L’Analyse retire d’une main ce qu’elle donne de l’autre. Je me détourne eveceffroi et horreur de cette plaie lamentable des fonctions continues qui n’ont point dedérivées et je viens vous féliciter bien vivement de votre merveilleux développement enfraction continue de l’intégrale....”