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BEGINNINGLESS PAST ENDLESS FUTURE AND THE ACTUAL INFINITE Wes Morriston One of the principal lines of argument deployed by the friends of the kal m 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 endless series of events is possible must therefore reject this popular line of argument against the possibility of an actual infinite Advocates of the kal m cosmological argument claim that a series of events in time must have a beginning Putting the matter a bit more precisely 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 beginningless 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 kal m cosmological argument 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 follows that something must be wrong with this entire line of argument I shall begin by briefly reminding the reader of how the general argument 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 an FAITH AND PHILOSOPHY Vol 27 No 4 October 2010 All rights reserved 439 Faith and Philosophy 440 endless future which is clearly possible I will argue that no relevant distinction 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 appeals to intuitions that he expects most people to share One of his favorite examples is that of Hilbert s Hotel The in famous hotel has infinitely many rooms each of which accommodates exactly one guest Each is occupied 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 1 Craig 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 metaphysically impossible it does not immediately follow that an infinite number of things cannot exist The allegedly absurd implications of a Hilbert s Hotel for example that infinitely many additional guests could be accommodated by creatively moving its guests around follow only because infinity 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 primarily 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 implications 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 1 Just in case anyone needs reminding one might make room for one new guest by moving 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 441 in time cannot Isn t it enough that we can mentally move the guests about Suppose their rooms had no doors 2 Craig 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 reproduce the worry about inverse arithmetical operations Even with no doors the absurdity of a Hilbert s Hotel would still be quite apparent Initially it may not be easy to see how this constitutes much of a response to the objection Doors can be created So the guests can still be moved and not only mentally It is metaphysically possible for the guests to be moved It is not