There are two types of infinity, potential and actual. Craig defends this premise with the argument, not that a potentially infinite number of things cannot exist, but that an actually infinite number of things cannot exist. This would defend the premise because in a beginningless universe, prior to the present event, there would have existed an actually infinite number of events. But, if an actually infinite number of things cannot exist, then the universe must be finite in the past, and have begun to exist. An actual infinity is the “number of all the numbers.” It is called the “transfinite number,” because it comes after all the finite numbers. “All the numbers” is literally all finite and distinct numbers in the series 1, 2, 3,. . . Since the actual infinity represents all the finite numbers as a set, no numbers can be added to it. Potential infinity, on the other hand, is simply growing towards infinity as a limit. But it never reaches actual infinity, for there is never a time when it becomes “full” and all numbers have been added to it, since one can always add another finite number to potential infinity. The “fullness” of actual infinity is potential infinity’s limit.
Though the actual infinite is an accepted concept within mathematics, it “is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought. . . . The role that remains for [it] to play is solely that of an idea” (Hilbert, “On the Infinite” in Philosophy of Mathematics 141). The impossibility of the actual infinite becomes apparent once it is transposed to the real world: counter-intuitive absurdities result. For example, an actual infinity of kilometers is the same as an actual infinity of meters. This is because, to reiterate, the actual infinity represents all the finite numbers as a set. If you measure out an actual infinity of kilometers, you have measured out all the finite numbers together as a distance, though you represent them through the measurement “kilometer.” Since actual infinity is all the finite numbers, it can’t be added to, and if it were subtracted from, it would no longer be an actual infinity, since it has to include all numbers. An actual infinity of meters, then, is still measuring out all the finite numbers together as distance, though represented through the measurement “meter.” The meters and the kilometers are two different measurements, but each is measured out to actual infinity. The amount a meter measures is different from a kilometer, being one-thousand for every kilometer measured. Once one has an actual infinity of meters, then, for every one meter that represents one of the finite numbers in the set of all finite numbers, you have one-thousand kilometers. The distances would have to end up different, and yet they must be the same if they are to be an actual infinity, because an actual infinity cannot grow any greater or lesser or it would cease to be an actual infinity. If an actually infinite number of things were possible, then such measurements are possible. And if such a measurement is impossible, then so is real existence of an actual infinite. It becomes a matter of what the more likely, more astronomically probable choice is.
One critic of this argument, Graham Oppy, chooses the less probable choice. He says—No! The actual infinite is possible—because this measurement is possible. His defense is as follows: “[T]hese allegedly absurd situations are just what one ought to expect if there were . . . physical infinities” (Oppy, Philosophical Perspectives on Infinity 48). Oppy doesn’t defend his belief that “absurd situations” and the “actual infinity” exist, he only admits that, yes, we should expect there to be absurd situations. Those who believe in the actual infinity, and those who don’t, both agree that such situations are the result of an actual infinity. But does this make the actual infinity any more likely? Or does it make the actual infinity less likely? Apparently neither, according to Oppy, since these situations are not likely anyway. And if they make the actual infinity neither more nor less likely, then both their and the actual infinity’s existence is a matter of personal preference. Oppy therefore admits that he believes in the actual infinite out of personal preference—though those who don’t believe would also be doing so out of personal preference. If it is really a matter of personal preference, Oppy needs to explain, at least, why he does believe the actual infinite exists in reality. Those who don’t believe, if out of personal preference, do so in preference of a less absurd and a more realistic world, as the mathematician Hilbert explains in Philosophy of Mathematics. Supposing, of course, that we have a pretty good idea of a “realistic world.” Oppy does give his reasons in the beginning of his own book, Philosophical Perspectives on Infinity: ". . . t must be possible to convince reasonable religious believers that traditional monotheistic arguments for the existence of God are worthless" (xi). One of these arguments being the kalam, a premise of which is that the universe began to exist. Oppy has a preference for a universe that did not begin to exist, because if it did exist, it could be a true premise for an argument for the existence of God. That argument would then no longer be worthless, since it would be valid and sound. But Oppy is here presupposing that the arguments are worthless, and therefore he has to believe in an actual infinity, or else consider a different view, one that is not his own and is not preferable. Not that the arguments aren’t worthless, but they are arguments for a more likely personal preference, given their premises. Oppy’s personal preference is not astronomically probable, because he has no arguments, and therefore no valid or sound ones.
