Here is another counterexample to the claim that it is impossible to form a concrete and actual infinity through successive addition. Consider a gamma-widget. A gamma-widget is a stochastic critter that, once it is produced, produces at least one offspring. How many offspring it produces and how quickly is random. For every n>0, a gamma-widget has probability 2−n of producing exactly 22n offspring in exactly 2−n years. The offspring are exact copies of the parent (this is a way in which the gamma-widget differs from the beta-widget). They, too, are gamma-widgets, and hence produce offspring. All the probabilities are independent.
Here is a mathematical fact (at least, it is a fact, if my scribbled calculations are right): There is a non-zero probability that if a gamma-widget comes into existence, it will be an ancestor of infinitely many gamma-widgets a year later. What has non-zero probability is also possible. Therefore, it is possible for this to happen. Here is another fact: The probability that there is a finite number n such that n years later there are infinitely many gamma-widgets is one. What is almost certain (i.e., has probability one) is a fortiori possible. Therefore, it is possible to have an infinite number of gamma-widgets arise through successive addition.
(If one is worried that the addition is not successive because multiple offspring are produced at the same time, we can stagger the productions in some way, and then not count the ones whose productions happen to overlap.)
Does this argument, and that in my previous post, destroy Craig's Kalaam argument? I think not yet. For there two ways that an infinity prima facie can be built up by successive addition. In one way, there is a first addition but no last one. In the other way, there is a last addition but not first one. What my examples show is that the first way is possible—there is a first addition but no last one. But I have not shown that there is no possibility of successive addition of the other sort.
However, the latter possibility follows from a modification of the case. Suppose that each gamma-widget that comes into existence before the year 2020 randomly and uniformly chooses a time during 2020, and completes a mug at that time. Then there is, almost surely, a subset of the mugs that has a last addition but no first one.
Craig can get out of these arguments by supposing that time cannot be infinitely divisible. Or he might just deny the possibility of my widgets. But what grounds would there be for that denial? Consider a variant on the gamma-widget, the delta-widget. It is just like the gamma-widget, except that it has probability 2−n of generating n2 offspring in 2−n years. The delta-widget is just like the gamma-widget, except it is much less prolific: n2 instead of 22n. I am pretty sure (I haven't checked) that with probability one, delta-widgets do not have an infinite population explosion. So Craig can't object, it seems, to delta-widgets except by denying the infinite divisibility of time. But if delta-widgets are possible, why not the more prolific gamma-widgets?
Maybe my calculations are wrong. But I am pretty sure they're not. If they are wrong, replacing 22n with an even faster growing sequence should fix things.