Last month’s multiverse discussion focused on one of its less controversial aspects—the idea that the universe extends beyond the limits of our observations. The uniformity we see in our universe (the cosmic microwave background radiation being the best example) strongly argues for this point. The issue then becomes how large the actual universe is. Using the maximum curvature detected by WMAP and a simple assumption that the universe closes back on itself, a minimum size for the whole universe roughly equals 1000 times the size of the observable universe. (For a discussion of these terms see my initial multiverse article.)
Somewhat more controversial is the idea of a spatially infinite universe—a result that derives from the current formulations of how inflation works. As discussed last month, a spatially infinite universe dramatically impacts the apologetic significance of some finetuning arguments, although it would still comfortably fit within a Christian worldview.
As it currently stands, any experimental verification of inflation’s details lay far in the future so the conclusion of a spatially infinite universe remains a more philosophical issue at this point. In such light, I thought it relevant to highlight some philosophical arguments against a spatially infinite universe advanced by William Lane Craig. Craig uses them to support the Kalam cosmological argument but they apply here as well.
An article in Scientific American gives more details of the relevant science, but the point pertinent to this discussion involves a transformation of the infinite future expansion of our bubble into an actual spatially infinite universe. Craig argues that actual infinities of the type invoked here cannot exist because they lead to absurdities.
He outlines a few examples of absurdities arising in dynamical infinities in this article published in the Canadian Journal of Philosophy:

Consider an infinite hotel full of guests. Now suppose another infinite group arrives and asks for rooms. If the owner has each guest move to the room twice their current value (1 to 2, 2 to 4, 3 to 6,…), this leaves open the infinite number of oddnumbered rooms. So a completely full hotel can accommodate an infinite number of new guests.

Consider two planets where one orbits twice as fast as the other. After an infinite time, each planet has accumulated an identical number (the infinite value alephnull, ) of orbits. However, during every possible finite time interval, the faster planet accumulates twice as many orbits as the slower.

In the previous example, one could ask the question of whether the number of completed orbits is even or odd. After an infinite time the number of orbits is a value referred to as alephnull. An even number is a multiple of two; an odd number is one more than a multiple of 2. But, = (2 x ) = (2 x + 1). So the number of orbits after infinite time is both odd and even.
These examples highlight that basic rules which we take for granted cannot apply in physically existing infinites. Either we must rewrite basic arithmetic rules (addition, subtraction, multiplication, division, and comparison) or such infinities do not exist. I have glossed over many details, but the objections Craig raises are worth a serious look as a response to infinite, dynamic universes. Additionally, scientists typically regard infinities as a sign that they have entered a region where their theories are no longer valid.
So, there may be good philosophical reasons to reject the notion that we live in a spatially infinite universe. Even so, the issue of the actual spatial extent of our universe still remains. It could still be so large as to negate the apologetical significance of the finetuning arguments. However, that scenario also poses some significant philosophical issues that I will address next time.