What exactly does it mean to be ‘infinite?’ I know we often use the term for rhetorical flourish (usually in hyperbole or embellishment), but what precisely does it mean? The other night a friend at the Dallas Socratic Society discussion group, Brett, read a fantastic paper, Examining Infinite Set Possibilities. Brett’s aim was simply to sketch out some coherent ways we can speak about infinity and infinite sets. Aristotle long ago, Brett reminded us, gave us the traditional view of infinity:
…it is always possible to think of a larger number; for the number of times a magnitude can be bisected is infinite. Hence the infinite is potential,never actual; the number of parts that can be taken always surpasses any assigned number. Physics 207 b8
So if Aristotle is right, an actual infinite set is just impossible because one can always add another member to the set. But if this is true, then the set is always finite. William Craig uses the notion of the impossibility of an actual infinite to bolster the second premise in the Kalam argument. The Kalam argument, recall, makes the following move:
1. Whatever begins to exist has a cause
2. The universe began to exist
3. Therefore, the universe has a cause
Craig supports (2) by (among other things) arguing for the impossibility of an actual infinite series of events in time:
4. The series of events in time is a collection formed by adding one member after another.
5. A collection formed by adding one member after another cannot be actually infinite.
6. Therefore, the series of events in time cannot be actually infinite.
I should say that Carl Sagan may have been a smart cosmologist, but this argument makes his proclaimation about the universe being “all that ever was, is, or will be,” appear quite silly.
Now while I readily accept the notion that an actual infinite set consisting of concrete individuals is just impossible, the Platonist in me is quite willing to accept the notion of an actual infinite set of abstract objects. After voicing this in the discussion, another friend Sloan—who I should say is far smarter than I—roundly criticized me for embracing this view. But I can’t shake it. It seems to me that sets containing members like possible worlds, numbers and other abstracta, are actually infinite! What’s wrong with me? Am I just a Platonist gone wild!?