How God could create a Hilbert’s Hotel by successive addition
March 30, 2009 — 10:00

Author: Wes Morriston  Category: Uncategorized  Comments: 34

I want to throw this little argument out for comment. (This type of argument was first suggested to me in correspondence with Mike Almeida. My colleague, Michael Tooley, thinks that an argument along these lines is sound.)
The argument implicitly assumes two things.

1. Time is infinitely divisible.

2. There is no least amount of time that it would take God to create a hotel room.

If these assumptions are granted, then it seems that God could not only create a Hilbert’s Hotel, but could do so by successive addition. As I’ll describe the scenario, God does it in two hours.
During the first hour, God creates the first room. During the next half hour, He creates the second room, during the next fifteen minutes, He creates the third room, during the next seven and a half minutes, He creates the fourth. He continues in this manner until two hours have elapsed. At that point, God has created infinitely many rooms.
We’re (obviously) dealing with an actual infinite here, since the two hours have elapsed, and all the rooms have been created. So it looks as if an actually infinite number of “room creations” have taken place and an actually infinite number of rooms exist at the end of the two hours.
This sort of scenario has a familiar air of Zeno-like paradox. Add the following twist on it to bring out the weirdness. Suppose that there’s a switch with just two positions: ON and OFF. At the start of the process, the switch is in the OFF position. Every time God creates a room, He changes the position of the switch. Will the switch be ON or OFF at the end of the process?


This question is not answerable, given merely what’s specified in the scenario as I have described it. The infinite series of switchings does not entail either that the switch will be ON or that it will be OFF. It could be either.
Does this show that the whole scenario is impossible?
In his discussion of “supertasks” in an appendix to The Kalam Cosmological Argument, Craig discusses scenarios like this one, and declares them to be impossible. Why? Because the position of the switch at the end of the two hours must be causally determined by what went on during the two hours. The only way for that to be so is for there to have been an “infinitieth” change in the position of the switch. But there can’t be any such number. At any point during the series of switchings the number that have occurred is finite.
I am not satisfied with this response. What does “must” mean in “must be causally determined by what went on during the two hours?” In order to show that the scenario involving God is impossible, it must have the force of metaphysical necessity. So understood, I don’t think Craig’s claim is true.
Craig’s claim might be true if we were dealing with a merely natural process (in which, among other things, nothing can travel faster than the speed of light). But a God who can create the entire universe out of nothing need not be bound by the causal rules that hold in the world He has created. So I don’t see why God couldn’t simply settle the matter. If God so chooses, the switch is in the ON position at the end of the process. If God so chooses, it is in the OFF position. If God so chooses, the switch ceases to exist altogether. The fact that the scenario by itself doesn’t tell us which position the switch will be in does not therefore show that the scenario metaphysically impossible.
As far as I can see, the only way to block the argument as I’ve stated it is to reject the claim that time is infinitely divisible. If, for example, you thought that any finite chunk of duration was necessarily made up of finitely many temporal atoms each of which comes into being all at once and ceases to be all at once, then the scenario would be impossible. Even an omnipotent God would run out of sets of temporal atoms in which to create hotel rooms.
I’m not a philosopher of time, but it does seem to me that this solution isn’t very plausible. Others may have more sophisticated thoughts.

Comments:
  • Mike Almeida

    The only way for that to be so is for there to have been an “infinitieth” change in the position of the switch. But there can’t be any such number. At any point during the series of switchings the number that have occurred is finite.
    Wes, there is a simple counterexmaple to this. Let C be a perfectly accurate clock, or the closest we have to that. Let C be wired so that it causes P to explode after the instant at 8am, but no time before or at 8am. As the hand of C sweeps past 8am, there is no earliest moment at which it detonates P, since there is no earliest moment at which the hand of C arrives after 8am (for each possible instant after 8am, there are infinitely many earlier and earlier moments). Yet there is no doubt that it can be so set that it detonates P after 8am and at no particular time after 8am.

    March 30, 2009 — 11:35
  • MHart

    Wes,
    My intuitions are with Craig about the impossibility of supertasks. I’m not sure that God’s settling the matter helps with the problem, for the question then seems to become: “Was the switch on or off before God chose to settle the matter?” Or better, “Had God not decided to settle the matter, would the switch be on or off?” So a lingering suggestion of metaphysical impossibility seems to remain.

    March 30, 2009 — 11:37
  • 1. I agree with you on the inadequacy of the Thomson’s Lamp arguments. The position of the switch at the end of the process is underdetermined by the description of the process, but we have been given no reason to think it should be determined by the description of the process.
    2. But as to your main argument, why can’t Craig say this:
    (*) Any stretch of the process starting at the beginning but ending short of the two hour mark is possible, but the process as a whole is not possible.
    There are, after all, processes satisfying (*). For instance, consider the following process. During the first hour, God says: “I promise that next year I will create finitely many horses, at least one in number.” During the next half hour, God says: “I promise that next year I will create finitely many horses, at least two in number.” During the next 15 minutes, God says: “I promise that next year I will create finitely many horses, at least three in number.” And so on. The process truncated at any length of time short of 2 hours is possible. But the process going for the whole two hours is impossible, since then God would have made an incompossible set of promises (to satisfy all the promises, he’d have to create a finite number of horses which exceeds every integer), and an essentially morally upright God can’t do that.
    3. By very similar kinds of reasoning to yours one can generate Grim Reaper situations.
    4. Still, I think there is a lot of plausibility to the thought experiment, and I do not know that Craig can account for all of this plausibility (say by positing a confusion between the possibility of the process running for every length of time short of 2hrs and the fact that for every length of time short of 2hrs the process can run for that).

    March 30, 2009 — 11:54
  • Sandy

    I worry that having God decide the position of the switch is cheating out of the spirit of the paradox. To block it one might simply propose that God subcontracts: Suppose that God creates a being whose only purpose or capability is to make hotel rooms; time is infinitely divisible; and there is no least amount of time that it takes God’s subcontractor to create a hotel room. We attach a switch, run the Zeno-like sequence, and the subcontractor comes back after two hours, satisfied with a job well done. In what position does he find the switch?

    March 30, 2009 — 12:01
  • Anonymous

    Given that Craig thinks there are no actual infinities, I doubt he can consistently hold that time is infinitely divisible: if it were, an hour would be composed of an infinite number of instants, but an infinite number of instants is an actual infinity. If he did concede that time is infinitely divisible, the a priori horn of his Kalam argument would lose its force.
    I suspect that orthodox theism requires a commitment to at least one actual infinity anyway. If God is omniscient, he must know that 2+2=5 is false, 2+2=6 is false, 2+2=7 is false, ad infinitum. If God knows that 2+2=5 is false, then God must believe 2+2=5 is false, and the same for the rest. But then God has an infinite number of beliefs. Craig might try to replace omniscience with some kind of potential omniscience, wherein God has the capacity to know all truths and falsehoods without knowing all of them at any given point in time, but this would constitute quite a deviation.
    Perhaps the conjunction of the bizarreness of temporal finitism and potential omniscience constitutes a defeater for Craig’s arguments against the possibility of actual infinities? I’m not sure, but they at least give us reason to scrutinize them closely.

    March 30, 2009 — 12:17
  • Wes Morriston

    Thanks. Just a brief comment on point #2 in Alexander Pruss’s post…
    I agree that in the scenario described there, God cannot keep all the promises made during the two hour period. So if it’s metaphysically impossible for God to break any promises, then the scenario is also metaphysically impossible. But it seems to me that this shows only that some “supertasks” are impossible – not that all are.
    Is there a problem with saying that some supertasks are possible and others aren’t?
    (When I have time, I’ll look into the Grim Reaper situations.)

    March 30, 2009 — 17:33
  • Wes Morriston

    Reply to MHart:
    Let’s agree that one of the following propositions must be true:

    1. At the end of the two hour period, the switch will be in the ON position.
    2. At the end of the two hour period, the switch will be in the OFF position.

    God could settle the matter, but you are quite right to say that He need not do so. This leads you to ask: Would the switch be ON or would it be OFF if God did not act?
    Neither answer seems right. But I’m not sure why this question has to have a definite answer. Granted that the switch WILL be on or WILL be off, it does not follow that it WOULD be on or WOULD be off. We might want to say that both subjunctive conditionals are false, or we might instead want to say that they lack any definite truth value.
    Bringing God in helps make it seem possible for the switch to be in whatever state it is in at the end of the two hour period by providing a possible cause. But this does not – and was not meant to – provide any ground for supposing that the subjunctive conditional, “It WOULD be on (or off) given what what goes on during that two hour period” is true.
    Sometimes, if we want to speak the sober truth, we have to settle for “it would probably be the case that…” or “it might or might not be the case that…” That seems to me to be the way things are in the case as I have imagined it.

    March 30, 2009 — 18:27
  • Anonymous no. 2

    Anonymous,
    1.
    I believe Craig thinks that segments of time are potentially infinite without being actually infinite. That is, segments of time can always be further subdivided, but they do not actually consist of infinitely many temporal points (or events or subsegments or whatever).
    I’m not sure if this position makes sense (time consists of finitely many temporal atoms, and yet these atoms can always be further subdivided?), but that’s what he says about the divisibility of space, and so I’m guessing that’s what he’d say about time.
    2.
    Perhaps Craig can appeal to divine simplicity, holding that each of God’s beliefs is identical with each other of God’s beliefs. Perhaps the view would be that, though there are infinitely many beliefs as individuated by what is conceptually distinct, there is only one belief (or perhaps just the divine intellect itself) as individuated by what is really distinct, and that only what is really distinct falls under the ban on actual infinites. But I don’t know the ins and outs of simplicity, so this is just a guess at how this might go.

    March 30, 2009 — 18:42
  • Wes:
    I agree that my example only shows that some supertasks are impossible. The point of the example was to show that all the finite cases can be possible, with the infinite not possible.
    The Grim Reaper paradox is another thing to show this. It can be put like this. A Grim Reaper (GR) is a being that with an exact alarm set to a particular time. When a GR’s alarm goes off at t, it checks whether Fred is alive or not. If Fred is alive, it kills Fred, instantaneously at t. If Fred is not alive, the GR does nothing. The killings are permanent. It seems a GR is a possible being.
    Suppose there can be infinitely many beings. Then, plausibly, there can be infinitely many GRs: G1, G2, ….
    Suppose that for each n, Gn’s alarm is set for 1/n seconds after midnight. If one may posit super tasks where things happen at an infinite set of times, this seems possible. Suppose also that Fred is alive at midnight, and there is nothing to kill him other than the GRs.
    Assume now that t is any time after midnight. Then Fred is dead at t. For if Fred were alive at t, then he was alive for all of the interval [midnight,t], and infinitely many of the GRs activated on that interval, and Fred cannot survive the activation of a GR. So, Fred is dead at every time later than midnight. But no GR ever actually does anything other than scan Fred. For at any given time after midnight, Fred is already dead. Who killed Fred?
    If you worry about instantaneous activation, we can modify the story slightly. The GR that activates at midnight+1/n seconds takes 1/(4n) seconds to scan whether Fred is alive and if he is, it takes another 1/(4n) seconds to kill Fred. The story still works.
    Here is a somewhat more complex version of the story.
    Anonymous:
    Wes’s argument does not assume that time is infinitely subdivided, but only that it is infinitely subdividable. Thus, between 1:00:00.0 pm and 1:00:00.1 pm, there might not actually be a time (such as 1:00:00.05), but there can be–there is a possible world in which there is such a time.
    Basically, Wes is trying to show that once one admits that each interval of time is infinitely subdividable, one should also admit that it is possible to have an interval of times that is infinitely subdivided. This sounds trivial, but if one keeps track of the order of quantifiers, it is far from trivial.

    March 30, 2009 — 19:39
  • In this scenario, how does one know when exactly to stop creating rooms? How does the two hours “arrive” it seems you have to keep dividing and will always have time left…so it is a paradox and hence I do not see the actual infinite being produced…only the potential for infinite room creation. Though the two hours of time is actually being “whittled down” the creation is still proceeding temporally into a potentially infinite future. Of course, I could be wrong…

    March 30, 2009 — 19:40
  • To expand slightly on Wes’s comments about the switch, the question of what position the switch would be in seems to me to be just a case of an underspecified counterfactual. There are lots of those:
    1. If the moon were made of cheese, what kind of cheese would that be?
    2. If an engineer made an on-off switch, would it first be on or first off?
    3. If x were a human being, would x be male or female?
    4. If y were a number, would it be even or odd?
    To all of these, the answer is something like “it depends on further details of the situation” or “either one”.
    Sometimes it seems like there is an answer when we put the question in story form, “suppositionally.” But there, too, there need be no answer: “Suppose Jordan drove too fast, and got stopped by the police. Jordan’s father is a congressman, and so the police let Jordan off. Is Jordan a man or a woman?” Of course, the right answer is: “The story doesn’t say!” (If one says that men are more likely to drive too fast than women, then one still cannot say either way, but one should say: “More likely a man than a woman.” Besides, the story may have been told precisely to challenge stereotypes–sampling in the class of stories is different from real-world sampling.)

    March 30, 2009 — 19:50
  • Wes Morriston

    After each subdivision and its corresponding creative act, God has created only finitely many hotel rooms, and has “time left” (within the two hour period) to create more.
    The fact that God steadily increases the speed with which He is creating hotel rooms makes no difference to the passage of time. The two hour period must come to an end. When those two hours have elapsed, infinitely many hotel rooms have been created, and God is “finished” with this particular “supertask.”
    I know it’s weird, but it will seem less so if you abandon a certain “picture” of what’s going on here. Don’t think of the task to be performed as having n parts, and of God as being “finished” when He has completed the nth part.
    BTW, God doesn’t have to “stop.” He can keep right on creating hotel rooms when the two hour period has elapsed. But that’s perfectly consistent with saying that he has already – during the initial two hour period – created infinitely many of them.

    March 30, 2009 — 21:23
  • Wes:
    Is there any way you can package your construction into a valid argument, say based on some principle of (re)combination? Or does it inextricably depend on a reaction of “yes, that sounds possible”?

    March 30, 2009 — 21:59
  • Anonymous

    Alexander Pruss:
    I’m inclined to think that, at least in this case, being infinitely subdividable will result in being infinitely subdivided.
    Consider your proposal:
    (*) Any stretch of the process starting at the beginning but ending short of the two hour mark is possible, but the process as a whole is not possible.
    This has the following consequence:
    (C) For any finite amount of time in the two hour period, there is a possible world in which that finite period of time is subdivided.
    As far as I can see, there are two possibilities, temporal atoms or instants. Some temporal atom theories might think the atoms are large, say half a second; others think them very small. Yet no matter how small the proposed temporal atom is, there is a possible world in which the duration of that temporal atom is subdivided. Then all temporal atom theories are false. But if all temporal atom theories are false, then time must be composed of instants, at least in one possible world. So we again have an actual infinite.
    Perhaps you can see a way out of this dilemma. For my part, on reflection I don’t quite see what’s so absurd about thinking there are temporal atoms. It seems to me that a lot of Zeno’s paradoxes can be recast toward time; one solution to them regarding space has been to hold it is discrete, so why not time? Nor does the kind of potential omniscience I brought up strike me as being quite as peculiar as I thought it was.

    March 30, 2009 — 23:47
  • I think then I would have to reject premise 1 when taken together with events happening over a span of finite time (ie within a 2 hour time limit). I think this is why we have to say that God always has “time left” but in actuality, if the two hours is “up” then there is no real time left. This is a thought experiment using a potential infinite thinking about time as unbounded set. Time only exists as such potentially in the future…not in the past.
    So it seems premise 1) creates a contradiction (paradox if we like) only when attempting traverse time that can expire – finite time. So if the sands in the hour glass can actually run out, then there are no correlated events of creation that continue. I am assuming your experiment is grounded in that these creative acts take place correlated to and in finite time (of some possible world). It is my understanding, that this is why Craig argues that we could not traverse an actual infinite by successive addition.
    Going into the future, one might create a very large, unbounded set of rooms, yet if the clock is going to be stopped, then there are simply no time/creation-event correlations remaining and the hotel rooms could be all counted just as a finite number of events past can be traversed.
    But I can take your word for it for now and keep reading others’ comments – it is quite a fascinating paradox.

    March 30, 2009 — 23:49
  • Wes Morriston

    Consider the following:
    (A) Any stretch of the process starting at the beginning but ending short of the two hour mark is possible.
    (B) The process as a whole is possible.
    Are you looking for premises that will get us from (A) to (B) for the process as I’ve described it?
    I’m not good at the formal stuff. Maybe you could come up with something? Or do you suspect that it simply can’t be done?

    March 31, 2009 — 8:10
  • Anonymous:
    “As far as I can see, there are two possibilities, temporal atoms or instants.” A third option is that time is made up of intervals which in fact have no parts (like atoms) but which could have had parts (unlike atoms). This third option is Aristotle’s.
    Wes:
    I am suspecting that it’s going to be really hard to come up with a non-question-begging strategy for getting from (A) to (B), especially given my promise counterexample which shows that the inference from (A) to (B) does not hold in all cases.
    Maybe this will do something, though. Let A(n) be a finite or infinite sequence of positive lengths of time. Let B(n) = A(1)+A(2)+…+A(n). Say that the sequence A is Hilbertian iff possibly[(for all n)(God creates a new hotel room at B(n))].
    Craig, I think, is then committed to the following claims:
    (1) If A is a finite sequence, then A is Hilbertian.
    (2) If A is an infinite sequence such that B(n) tends to infinity as n tends to infinity, then A is Hilbertian. (Consider this special case: A(n)=1 for every n. Then B(n)=n. And God could create a new hotel room at time n for every n.)
    (3) If A is an infinite sequence such that B(n) tends to some finite number as n tends to infinity, then A is not Hilbertian.
    So, Craig is committed to saying that God can create a sequence of hotel rooms at every B(n) iff it is not the case that B(n) converges to a finite number as n tends to infinity. (B(n) is an increasing sequence, so it must either tend to infinity or to a finite number by a standard theorem of real analysis.) But why should it matter whether B(n) converges or not for the question whether God can create such a sequence of hotel rooms?
    Consider these two sequences A(n):
    (a) A(n) = 1/n
    (b) A(n) = 1/n^2
    Since in case (a), B(n) tends to infinity (1+1/2+1/3+… is infinity), and in case (b), B(n) tends to a finite number (1+1/4+1/9+… is finite), Craig seems to be committed to saying that the sequence in (a) is Hilbertian while that in (b) is not. But this seems a really strange limitation on divine power.

    March 31, 2009 — 8:44
  • MHart

    Wes,
    So if you grant that God doesn’t have to intervene and that the end state of the switch can be the result of the preceding natural causal process and nothing more, then we get the conclusion that, M-possibly, supertasks are absurd. But isn’t that all Craig needs to motivate his rejection of the actual infinite? I don’t see why he needs the stronger claim that, necessarily, every supertask instance results in the relevant absurdity, merely that it is possible that (at least) one does.

    March 31, 2009 — 10:53
  • Wes Morriston

    But I did not grant “that the end state of the switch can be the result of the preceding natural causal process and nothing more.” On the contrary, I don’t see how it could be the “result” of such a two hour process. But that doesn’t worry me.
    In this connection, it’s important to see that there need be no fact of the matter about which state the switch “would” be in given the preceding two hours of divine activity. That was the point of making a sharp distinction between what does or will happen and what would happen.
    For a useful take on this issue, see Alexander Pruss’s post earlier in this thread on “underspecified counterfactuals.”

    March 31, 2009 — 11:22
  • Wes Morriston

    Thanks much. It will take me a little while to think this one through.

    March 31, 2009 — 11:23
  • Wes Morriston

    Oops! My last post was meant to be concerned with Alexander Pruss’s previous post about “Hilbertian sequences.” Sorry about that.

    March 31, 2009 — 11:26
  • Wes Morriston

    You wrote:
    —————————————-
    Consider these two sequences A(n):
    (a) A(n) = 1/n
    (b) A(n) = 1/n^2
    Since in case (a), B(n) tends to infinity (1+1/2+1/3+… is infinity), and in case (b), B(n) tends to a finite number (1+1/4+1/9+… is finite), Craig seems to be committed to saying that the sequence in (a) is Hilbertian while that in (b) is not. But this seems a really strange limitation on divine power.
    —————————————-
    Very strange indeed… Notice that (b) is a subset of (a). If God could create a hotel room for each member of (a), it’s hard to see why He couldn’t create one for each member of (b).

    March 31, 2009 — 22:44
  • Wes Morriston

    The two hours in my original scenario will elapse. Now consider the series:

    1 hr, 1/2 hr,, 1/4 hr., 1/8 hr. …

    Unless you think that time is made up of indivisible temporal atoms of a finite size, this series is infinite. Nevertheless, the sum of the series is just 2 hrs.

    Craig has to say two things about this:
    1. For any interval in this series, God can have created a hotel room during that interval and during every previous interval in the series.
    2. But at the end of two hours, God cannot have created a hotel room during each of the intervals in the series.
    This has the following implication.

    At any time before the two hours is up, God can keep on creating hotel rooms during the intervals in the series. Nevertheless He has to stop before two hours have elapsed.

    That seems far weirder to me than a Hilbert’s Hotel. What forces God to stop?

    March 31, 2009 — 23:38
  • Anonymous no. 2

    Alexander Pruss wrote:
    “A third option is that time is made up of intervals which in fact have no parts (like atoms) but which could have had parts (unlike atoms). This third option is Aristotle’s.”
    So whether an interval has parts does not enter into its individuation, and is not intrinsic to it? That is, there can be an interval in one world identical to an interval in another world, despite their differing in whether they have parts, and the intrinsic facts about an interval fail to settle whether it has parts?
    I find this option very difficult to understand.

    April 1, 2009 — 6:11
  • Wes:
    “What forces God to stop?” That question is a great way of making the argument plausible. But what if Craig answers: “Logic”?
    Here, by the way, is a variant on your argument. Say that a machine x is a halfdget iff as soon as x is made, x makes a copy of itself in half the time that it took for x to be originally made. It seems that it should be logically possible that someone makes a halfdget. But if it were to happen, then in another time interval of the length it took to make the original halfdget, there will be infinitely many halfdgets.

    April 1, 2009 — 9:01
  • Wes Morriston

    Nice example. So what would stop God from making a halfdget?
    What if Craig answers: “Logic”? Without elaboration, that’s not much of an answer. It’s not as if Craig has uncovered a logical inconsistency in the concept of a halfdget (or in the successive creation of hotel rooms in my original example). Nor does he claim to have done so.
    Nevertheless, Craig’s claim has always been that such things are impossible in the “broadly logical” or “metaphysical” sense, so that even an omnipotent being could not create them. (And he accuses his opponent of begging the questiion if he argues, “God could do X; therefore X is metaphysically impossible.”)
    Now I admit that I have been unable to think of any non-question-begging argument for the conclusion that things like this are possible. With regard to some of these scenarios (for example, the Grim Reaper, or perhaps your promise-making example), it seems to me that the scenario generates a clear absurdity, or even a contradiction. But other scenarios involving infinitely many successive steps do not appear to do so.
    So I’m left with this question. What’s wrong with saying that some of them are metaphysically impossible, while others are not? Is there way for Craig to defend the claim that it has to be “all or none” here? Can he (or anyone) give a serious argument for that claim?
    Like everybody else, I hate it when philosophers pass the “burden of proof” back and forth – especially when both of them claim to know the truth of the matter. It’s a bit like a silly game of “hot potato.”
    Nevertheless… A large part of Craig’s career has been based on claim to be able to settle questions like this one. So it does not seem unfair to demand that he (and those who agree with him) make a case that consists in more than just saying, “But that’s metaphysically impossible” whenever they need to.

    April 1, 2009 — 10:22
  • I think I agree.
    Here’s a further note on the halfdgets. Say that x is a procrasting halfdget (phd, for short) provided that after being created, x waits a year doing nothing, and then makes a copy of itself in half the time that x itself was made.
    If an initial phd is made in a month, then in a year that phd will take half a month to make a phd, and a year later, the next phd will take a quarter of a month to make a phd, and so on. Since the phds are always spaced out by more than a year, Craig can’t say that logic prevents God from making a phd. But now we have something really weird. For it seems no harder to make a being that makes a copy of itself immediately in half time and then sits around for a year doing nothing (we may supplement the definition of a halfdget with that proviso) than it is to make a being that first sits around for a year doing nothing and then makes a copy of itself in half time.

    April 1, 2009 — 11:31
  • Ryan

    For what it’s worth, I’ve excerpted and linked to two lectures available on Craig’s website that likely indicate how he’d respond to Wes’s argument. The first is Craig’s discussion of the potential infinite v. the actual infinite. Given this statement, I think he’d claim Wes’s argument is essentially an argument from a potential infinite (which is exactly what he does in his recent newsletter). If that is true, then Wes’s argument does not address Craig’s, because Craig is referring to *actual* infinites (represented by the Hebrew character aleph) instead of merely potential infinites (represented by the horizontal “lazy eight”). The second lecture relates to God’s ability to perform supertasks and its relationship to the impossibility of *forming* an actual infinite by successive addition. The first lecture shows that Wes’s argument is based on a potential infinite
    and, therefore, simply doesn’t address Craig’s argument. Thus, Craig would likely say, that Wes’s (1) is referring to a potential infinite, not an actual one. The second lecture shows that if God could create an infinite universe, He couldn’t do so by successive addition, pace Wes. God would have to create the actual infinite (hotels, universes, whatever) all at once.
    N.B.: Craig’s quotes are my transcriptions of two podcasts available on his website. I’ve given a direct cite to the podcast along with citations to the times within the podcast itself. I’ve quoted lengthy sections to save your time from going there, and cited them to allow you to verify them.
    1. Wes’s argument appears to confuse the potential infinite with the actual infinite. This matters, because the kalam is perfectly fine with potential infinites existing. Instead, it argues against (only) actual infinites. Here’s Craig:
    “[7:40] A potential infinite is a collection which is at every point finite, but always growing toward infinity as a limit. So that this collection is really an indefinite collection: it is finite at every point in time you pick, but it grows ceaselessly forever. And, therefore, it grows toward infinity as a limit. It never arrives at infinity, but it approaches infinity endlessly. And so, this is why it’s called potentially infinite: the infinite serves merely as a limit toward which the series grows, but it never arrives there. [8:20]…[8:29]Now, by contrast, an actual infinite is a collection…[in which] the number of members in the collection exceeds any natural number that you can think of…So it’s not growing toward it infinity, it *is* infinite…[9:28] So, when we say in the first philosophical argument that an actually infinite number of things cannot exist we are not denying the existence of a potential infinite. Potential infinites can exist. For example, the distance between any two points (such as on this podium), could be divided in half, and then in half again, and then in half again…on and on and on to infinity. But you would never arrive at infinity. The podium would never be actually divided into an infinite number of bits. Infinity would merely serve as the limit toward which you could endlessly divide.”[23:28] Now other times, people will [try to] refute this premise by saying, “Well, we can find examples of an actually infinite number of things in the real world. For example, they’ll say, “isn’t every finite distance between two points [or, I’d add, any finite span of time b/t two points, as Wes’s “2 hour limitation”] capable of being divided into ½, ¼, 1/8, 1/16, and so on out to infinity? Doesn’t that prove that in any finite distance [or span of time] there is an infinite number of sub-intervals with an infinite number of parts?” Well, I think that this is, again, a fallacious objection that confuses a potential infinite with an actual infinite. It’s true that you can continue to divide any finite distance [or span of time] as long as you want. You can keep going on and on and on dividing into smaller and smaller parts. But that doesn’t prove that you have an actually infinite number of parts already there. That just shows that the number of divisions is potentially infinite.”
    http://www.rfmedia.org/RF_audio_video/Defender_podcast20040509CosmologicalArgumentPart2.mp3
    Cosmological Argument (Pt. 2), at times internally cited.
    Wes uses the latter understanding when he says “[t]ime is infinitely divisible.” But that is a potential infinite, not an actual one. Thus, Wes’s argument is constructed on the basis of concept of the infinite that the kalam doesn’t address. Thus, under Wes’s argument, God would never arrive at an *actual* infinite no matter how long God created the hotel rooms, because the rooms would always be
    approaching infinity, but never arrive at it.
    2. It seems Craig thinks God could *could* perform supertasks (I’m getting this from his lecture). Craig just thinks that God *doesn’t*
    (I’m getting this from Mike Almeida’s claim in another thread on this site) Here?s Craig:
    “So, it seems to me that this idea of trying to form an infinite collection of things by successive addition is simply maladroit. It’s impossible. It cannot be done. And therefore in modern set theory, any notions of successive addition [have] been done away with. When [a] mathematician[] says that there are, say, an infinite number of negative numbers, he doesn’t imagine that these are posited successively. He just says that given the definition of membership, the membership of the set is given immediately; simultaneously, all of the members are given. But it would be absurd to try to give them one at a time. So if God were to create, say, an infinite number of marbles or an infinite number of books in a library, he would have to simply say, “Let there be.” Boom! And all of the infinite marbles or books would come into existence simultaneously. But it would be impossible to try to form an actually infinite number of books or marbles by adding one member at a time, one after another.”
    http://www.rfmedia.org/RF_audio_video/Defender_podcast/20040516CosmologicalArgumentPart3.mp3
    at times: 9:21 to 10:37

    April 1, 2009 — 21:52
  • I don’t think Craig can consistently say both:
    (A) God could create a simultaneous infinite number of objects.
    (B) An infinite number of objects cannot come into existence by successive addition.
    For imagine that God makes an infinite number of hens come into existence simultaneously (by (A)). Then there is nothing to prevent its being the case that for every n, the hen #n lays its egg at 7-(1/n) a.m. And if so, then the eggs would come into existence by successful addition (at 6 am, at 6:30 am, at 6:40 am, at 6:48 am, etc.)
    Suppose Craig denies that such careful coordinating of egg-laying times is possible. Instead, let us suppose that the hens just lay eggs in the usual way. Then, the egg laying times will be randomly distributed (not necessarily uniformly, but I assume with an absolutely continuous probability distribution; maybe it will be a Gaussian centered on the “standard” egg-laying time) over some interval of times during which infinitely many eggs will be laid. Assuming time is infinitely subdividable, the probability that some pair of eggs will be laid at exactly the same time is zero. So, almost surely, no pair of eggs will be laid at the same time. And, so, the collection of eggs laid during that interval grows one by one, and is eventually infinite.

    April 2, 2009 — 7:42
  • I posted a cleaned up version of my halfdget story on my blog.

    April 2, 2009 — 8:30
  • Ryan

    Alex,
    You conclude: “And, so, the collection of eggs laid during that interval grows one by one, and is eventually infinite.” I’m not sure I follow how the eggs will be “eventually infinite,” by which I take you to mean “actually infinite.” Here’s why. Taking your example, God creates an actually infinite number of hens. Those hens lay eggs. All the eggs are either simultaneously laid or they aren’t. If they are simultaneously laid, then an additional actual infinite has been created, though not by successive addition. Thus, that scenario isn’t a defeater for the proposition that an actual infinite can’t be formed by successive addition. But if the eggs are not laid at the same time (whether they are laid at specific times or random intervals), for every egg “n,” the egg laid after n will be (n + 1). Thus, for every egg laid at any time, there will always be another egg to lay, which egg can be designated with a natural number, namely n + 1. If that is true, then this second scenario constitutes merely a potential infinite because the set of eggs are finite at every point but always approaching infinity as a limit. I may be missing something here. I so, please correct me.

    April 2, 2009 — 9:16
  • Let N be the number of eggs laid before 7 a.m. The nth egg is laid at at 7-(1/n) a.m. Therefore, for any number n, N>n. But if N>n for every number n, then N is infinite.

    April 2, 2009 — 11:26
  • And here is a way to run my construction without each halfdget having to somehow keep track of how long it was created in.

    April 3, 2009 — 9:05
  • eric sotnak

    I know this argument doesn’t quite work, but since all this is such fun thinking about…:
    During the first hour, the switch is off. It is on for 30 minutes, and then off for 15, then on for 7.5 minutes and so on.
    In each pair of off-on alternations, the switch is off for twice as long as the switch is on.
    This never changes in the infinite progression. So, statistically, it seems that at the end of the two hours, the switch is twice as likely to be in the off position as in the on position.
    So, the switch is off.

    April 11, 2009 — 23:05