I don’t think it an overstatement to say that the concept of the infinite plays a key role in the philosophy of religion. There are at least two senses in which ‘infinite’ is used. First, ‘infinite’ is often used to mean maximal, as in God’s infinite power, knowledge, and goodness. Second, many arguments in the philosophy of religion discuss ‘infinite number’ or ‘infinitely many’. It is this second sense of the infinite that I focus on in this post. Here are two recent examples of this second sense of the infinite, from Prosblogion, with select quotes (and links to the full posts):
Grim Reapers vs. Uncaused Beginnings
By Joshua Rasmussen on February 4, 2013 10:47 AM
‘Another idea is that there is a problem with producing an actual infinite number of events. This third idea, if correct, would seem to block the grim reaper argument against uncaused beginnings. But it would reinforce the argument for a finite past.’
Infinite multiverse, fine-tuning and probability
By Alexander Pruss on March 6, 2013
‘So it seems that the only reasonable place to put the probability shift is when you find out that there are infinitely many Joneses who rolled a die.’
In these posts, we have two puzzles, one involves what happens if there are countably many grim reapers, the seconds puzzle involves probabilistic reasoning if there are countably many Joneses. Indeed, puzzles and paradoxes of the infinite are many. Yet few have stopped to ask the question: Which objects are the infinite natural numbers? Put another way: How should the finite natural numbers, numbers like 7 and 113, be extended into the infinite? Put still a third way: When someone says ‘there are an infinite number of grim reapers’ or ‘there are infinitely many Joneses’, what sort of structures should the person be referring to?
I simply assume that there are better and worse ways for concepts to carve up the world. I also assume that the more something walks and quacks like a duck, the more likely it is to be a duck. Then, infinite natural numbers in a nonstandard model of the reals behave very much like finite natural numbers, are so are the correct extension of the finite natural numbers into the infinite. When someone says ‘there is an infinite past consisting of an infinite number of days’, the days should be the structure of an infinite natural number. This blocks any sort of Grim Reaper problem. If there are infinitely many Joneses, this must be referring to an infinite number (in a nonstandard model) of Joneses, and then the reasoning becomes analogous to the finite case. Indeed, I suggest that correctly using ‘infinite number’ and ‘infinitely many’ blocks all paradoxes of the infinite. For more, see the papers here and here, and a video here.
It might be asked: ‘But why can’t we discuss days of the structure of omega-star, that is: …-3, -2, -1, 0? Surely this is an example of an infinite number!’ No, it isn’t. Certainly it is infinite, but it is not an infinite number. And, as Aristotle held, such an infinity is always potential, never determined and actual. It is inexhaustible. If there is an infinite past, it is the structure of an infinite natural number (which, it should be noted, has a beginning). Infinite natural numbers in a nonstandard model of the reals are actual and determined. For this distinction, as well as a test to determine whether something is a potential or actual infinity, see here.
If infinite numbers are going to play a key role in reasoning, it might be a good idea to first figure out what the infinite numbers are.