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

282 god and infinityline, the plane, the volume, the volume in four or more dimensions – have the samecardinal number, c. We can go further still. Cantor showed that one can construct awhole series of transfinite cardinal numbers starting with a 0 and a 1 and leading to a ωand beyond this to a ω+1 , a ω+2 ,...,a ω ω, a a0 , and so on. In fact, there is no end to thekinds of transfinite infinity we can construct. At the same time, these infinite sets sharean important feature with finite sets because, no matter how complex they seem, theyare conceivable by construction; hence, the reason for Cantor to call them “transfinite.”What lies beyond even the transfinite numbers? According to Cantor, lying beyondthe transfinites is “Absolute Infinity,” symbolized as . In one sense Absolute Infinityis inconceivable. Yet in another sense we can know something about , namely, we canknow something about its properties! Does this lead us into a contradiction, namely,that is both conceivable and inconceivable?To see that it is not a contradiction, consider the converse. Suppose that is asconceivable as the transfinites described earlier. Then there must be some propertyP that is exclusively a property of , a property in terms of which we can conceiveexclusively of . Now in order to make inconceivable, we merely have to stipulatethat every such property P is, in fact, shared by both and some transfinite ordinal:there is no property P that is unique to . This means that we can consistently assertthe conceivability and the inconceivability of . On the one hand, we can conceive of because each of its properties P is shared by some transfinite ordinal. Yet becauseof this, we can never differentiate completely from the transfinite ordinals, giventhat we can never describe as possessing a property P that it does not share withsome transfinite ordinal. In essence, we can never distinguish between and thetransfinite ordinals because of the fact that all its properties are shared with eachof the transfinites. We can never know if we are conceiving of and not sometransfinite ordinal. In short, we can never conceive of as unambiguously distinctfrom a transfinite ordinal. is thus inconceivable because can never be uniquelycharacterized or completely distinguished as distinct from some transfinite ordinal.Instead, the transfinite numbers lead endlessly toward Absolute Infinity but neverbegin to reach it, since lies, absolute and unapproachable, beyond all comprehension.This argument is often referred to as Cantor’s reflection principle, 21 and it led Cantorto set up a threefold distinction regarding the infinite:The actual infinite arises in three contexts: first when it is realized in the most completeform, in a fully independent other-worldly being, in Deo, where I call it the AbsoluteInfinite or simply Absolute; second when it occurs in the contingent, created world; thirdwhen the mind grasps it in abstract as a mathematical magnitude, number, or order type.I wish to mark a sharp contrast between the Absolute and what I call the Transfinite, thatis, the actual infinities of the last two sorts, which are clearly limited, subject to furtherincrease, and thus related to the finite. 2221 From Cantor (1980, p. 378). This translation is from Rucker (1983, p. 10).22 Before exploring the potential usefulness of Cantor’s ideas on infinity for theology, I must acknowledge andbriefly respond to two challenges: (1) Do mathematicians agree with Cantor that infinity is a consistent andcoherent concept in mathematics? (2) Is Cantor correct in believing that the world is actually infinite?Regarding the first challenge, some mathematicians accept Cantor’s articulation of the transfinites anddefend Platonic realism. Others point to antinomies in this work, including those of Burali-Forti, Cantor