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Endless Future

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FAITH AND PHILOSOPHYVol. 27 No. 4 October 2010 439All rights reservedBEGINNINGLESS PAST, ENDLESS FUTURE, AND THE ACTUAL INFINITEWes MorristonOne of the principal lines of argument deployed by the friends of the kalām cosmological argument against the possibility of a beginningless series of events is a quite general argument against the possibility of an actual infinite. The principal thesis of the present paper is that if this argument worked as advertised, parallel considerations would force us to conclude, not merely that a series of discrete, successive events must have a first member, but also that such a series must have a final member. Anyone who thinks that an end-less series of events is possible must therefore reject this popular line of argu-ment against the possibility of an actual infinite.Advocates of the kalām cosmological argument claim that a series of events in time must have a beginning. Putting the matter a bit more pre-cisely, they claim that a series of discrete, successive events must—as a matter of metaphysical necessity—have a first member. On the other hand, they have no problem at all with the suggestion that a series of events in time might have no final member—that it might never come to an end. Why the difference? The received explanation is that a beginning-less series of discrete and successive events would be an actual infinite, whereas an endless one would be a merely potential infinite. The actual infinite is said to have absurd implications that do not afflict a merely potential infinite.In the present paper, I reject this explanation. I claim that if the main line of argument deployed by the friends of the kalām cosmological argu-ment against the possibility of an actual infinite worked as advertised, it could be employed with equal effect to show that an endless series of events is also metaphysically impossible. Since (as almost all would agree) an endless series of discrete events is not metaphysically impossible, it fol-lows that something must be wrong with this entire line of argument.I shall begin by briefly reminding the reader of how the general argu-ment against the possibility of an actual infinite goes. I will then exhibit the implications of this argument for the possibility of an endless series of discrete and successive events. The heart of the paper will be concerned with William Lane Craig’s claims (i) that an endless series of events is a merely potential infinite, and (ii) that this establishes a relevant distinction between the beginningless past (which is supposedly impossible) and an440 Faith and Philosophyendless future (which is clearly possible). I will argue that no relevant dis-tinction has been established, and that Craig’s approach to this issue must be abandoned.What’s Wrong with the Actual Infinite?To show that an actual infinite is impossible in the real world, Craig ap-peals to intuitions that he expects most people to share. One of his favor-ite examples is that of Hilbert’s Hotel. The (in)famous hotel has infinitely many rooms, each of which accommodates exactly one guest. Each is oc-cupied, and the hotel is “full.” Nevertheless, by moving the occupants around in just the right way, new guests can be accommodated—indeed, infinitely many new guests can check into the hotel without making any of the original guests leave the hotel.1Craig also stresses the point that inverse arithmetical operations yield inconsistent results for cardinal infinities. This too can be illustrated by a thought experiment involving Hilbert’s Hotel. Suppose that infinitely many guests check out. Depending on which ones leave, the hotel may be left with either a finite number, or with an infinite number, of guests. Mathematicians deal with this problem by leaving subtraction for cardinal infinities undefined. “In the real world,” however, Craig insists that guests could leave a hotel, no matter what its size. So if Hilbert’s Hotel actually existed, we would be stuck with these inconsistent results.Craig blames all these allegedly absurd implications entirely on the fact that Hilbert’s Hotel is infinite. From this he thinks it follows that an actually infinite collection is metaphysically impossible. But consider the following objection. Even if one grants that a Hilbert’s Hotel is metaphysi-cally impossible, it does not immediately follow that an infinite number of things cannot exist. The allegedly absurd implications of a Hilbert’s Ho-tel—for example, that infinitely many additional guests could be accom-modated by creatively moving its guests around—follow only because in-finity is combined with another salient feature of the hotel—viz., the fact that its guests can be moved. But when we return to the case we are primar-ily interested in here—that of a beginningless series of events—there is a relevant disanalogy. Past events cannot be “moved out of” their respective temporal locations. How, then, are we supposed to derive absurd implica-tions from an infinite (because beginningless) series of past events parallel to those that can be exhibited in an infinite hotel accommodating infinitely many movable guests?Here is Craig’s answer:Your objection to the Hilbert’s Hotel illustration is that the alleged absurdity arises only from the fact that the guests can be moved about, whereas events 1Just in case anyone needs reminding, one might make room for one new guest by mov-ing each of the current guests to the room with her old number plus one; and one might make room for infinitely many new guests by having each guest move to the room with double her old room number.BEGINNINGLESS PAST, ENDLESS FUTURE, ACTUAL INFINITE 441in time cannot. Isn’t it enough that we can mentally move the guests about? Suppose their rooms had no doors.2Craig’s point appears to be that even if the guests couldn’t “really” be moved about, we could still do the thought experiments required to show that an actual infinite is absurd. We could still conceive of all the present guests occupying the even-numbered rooms (instead of their present ones), and of infinitely many additional guests occupying all the odd-numbered rooms. We could also conceive of alternative scenarios that would repro-duce the worry about inverse arithmetical operations. Even with no doors, the absurdity of a


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