I am not completely convinced by the following argument, but let’s try it.

Let p be a positive real number. A p-widget is a device that on its back has written down a positive integer (perhaps in very small numerals), and that is physically necessitated to behave as follows: As soon as a p-widget w is made, it makes a copy of itself–another p-widget. The amount of time it takes to make a copy of itself is n^{-p} years, where n is the number on w’s back. Moreover, while making the copy, w inscribes n+1 on the copy’s back (all within that n^{-p} year period). Finally, a p-widget does not perish once made.

So, once a p-widget comes into existence with 1 written on its back, it makes a copy of itself in 1^{-p} years and the copy has 2 written on its back. The copy then takes 2^{-p} years to make a copy, which has 3 written on its back. And so on. However, it seems that the enemies of supertasks and actual infinities should not object to a p-widget if p is less than or equal to 1. The reason for that is that if a p-widget is produced where p does not exceed 1, then although the production times for subsequent p-widgets do get smaller and smaller, nonetheless there is no supertask or actual infinity involved–at any given time, there are only finitely many p-widgets. The reason for that is that if p does not exceed 1, then the amount of time for infinitely many p-widgets to come into existence is 1^{-p}+2^{-p}+3^{-p}+… and this is equal to infinity if p is less than or equal to 1.

On the other hand, if p>1, then this infinite series adds up to a finite number, and so after a finite amount of time, there will be infinitely many p-widgets. For instance, if the first 2-widget has 1 written on its back, then there would be an infinite number of 2-widgets after pi^{2}/6 years. This, of course, the enemy of supertasks and actual infinities will claim to be impossible. So the initial difficulty for the enemy of supertasks and actual infinities is that a 0.9-widget and a 1-widget could be made, but a 1.1-widget cannot. That seems problematic–why should there be this intrinsic logical limit on how much faster new widgets can be produced?

But there is a further move I want to make. If the only objection is to supertasks and infinities, then the opponent of supertasks and actual infinities should not object to a world that contains a 2-widget–as long as God miraculously intervenes to stop the reproduction of 2-widgets before the pi^{2}/6 years are up. For if God does so intervene, then no paradox ensues.

Now imagine a world that contains only God and physical stuff and a time sequence lasting at least two years (the magic number pi^{2}/6 is approximately 1.64493), including initially a 2-widget with 1 written on its back. It is now metaphysically necessary that if such a world is actual, then God miraculously intervenes at some time in the first pi^{2}/6 years. But that seems really, really strange: Why would God be necessitated to miraculously intervene? There is something very odd about the answer: “He must intervene to prevent a supertask or actual infinity.” (One could perhaps imagine a case where an essentially omnibenevolent, omnipotentent and omniscient being would have to intervene to prevent an evil. But to prevent a supertask or an actual infinity?)

So what should the opponent of supertasks and actual infinities say? I think she has to deny the possibility of p-widgets for p>1 altogether–even in worlds where God prevents them from completing their infinite population explosion. But I do think it would be very strange if p-widgets were possible for p not exceeding 1, but became impossible for p exceeding 1. One can vary the scenarios to underline this point. Suppose one admits the possibility of a 1-widget. Now imagine a variant on the 1-widget, but where each new widget instead of having n+1 written on it, where n was the number on its parent, the new widget now has the number (n^{1/2}+1)^{2} written on it. It’s easy to check that these modified 1-widgets, when started off with a modified 1-widget with the number 1 written on it, experience an infinite population explosion in exactly the same amount of time that 2-widgets do. But why should it be possible for each widget to write down n+1 but not to write down (n^{1/2}+1)^{2}?

If this is right, then the opponent of supertasks and actual infinities needs to deny the possibility of any p-widgets at all. More generally, she has to deny the possibility of sequences of self-reproducing beings whose reproductive period approaches zero. Moreover, supplementing the present reasoning by my gamma-widget contruction, we see that she has to deny the possibility of self-reproducing beings with random reproductive periods that can be arbitrarily close to zero. The only way she can do this, I think, is to insist that for any being, there is a minimum length of time that that being’s reproduction must take.

I think this way out is not obviously absurd. If one takes a causal theory of time, then one just might think there is a logical relationship between the complexity of a causal effecting and the length of time that the effecting takes. On such a causal theory of time, the internal time for each p-widget to produce an offspring might be the same–because the same thing causally happens. However, this will only give one the conclusion that it is impossible to have an infinite number of p-widgets in finite internal time (the time internal to the widgets). But the opponents of supertasks and actual infinities may want the stronger result that it is impossible to have an infinite number of p-widgets produced in finite external time, and I don’t know if that is defensible.

Alex,

Have you seen Bill Craig’s response to your earlier Alpha/Beta widget thought experiment? Here’s the link:

http://www.reasonablefaith.org/site/PageServer?pagename=q_and_a