Wednesday, November 23, 2011

Forming an Actual Infinite By Successive Addition

Philosophers are divided on whether or not an actual infinite can exist and, secondly, on whether an actual infinite (if possible in and of itself) can be formed by successive addition. Karl Popper, for example, objects to Craig's argument that an actual infinite cannot be formed by successive addition. Popper states that the problem only arises if we assume that there is a point in the infinite past. However, there is no point in the infinite past, since infinity is not a number per se, but a set of numbers.

Even assuming that this objection is satisfactory, it does nothing to undermine the Aristotelian argument against an infinite regress of essentially ordered causes. Let's assume that the past is infinite. Keep in mind that the infinity of past events is still composed of finite periods of time. Now, for each finite period of time, a new regress of sustaining causes begins to be instantiated:

..., -5, -4, -3, -2, -1, 0

So, at -5 we have a new regress of sustaining causes; and the same is true for -4, -3, and so on.

Cn Cn Cn
C3 C3 C3
C2 C2 C2
C1 C1 C1
-5, -4, -3, . . .

Before -4 can begin its instantiation of sustaining causes, all of the sustaining causes of -5 must be instantiated. However, beginning with C1 of -5, the regress of sustaining causes cannot arrive at infinity, since we are in fact beginning with a point (more specifically, a cause). This means the Aristotelian argument is immune to Popper's criticism. After all, there is an obvious disparity between an infinite past and an infinite regress of sustaining causes. While the former does not have a first point, the latter most certainly does.