When I presented certain proofs of the existence of God to some atheists, I was challenged to provide reasons for why there cannot be an infinite regress, say, of causes.
This issue seems to be a sanity test: whether one can distinguish between the world of ideas or the ideal things and the real world.
For example, as I have mentioned, the eminent economist Steven Landsburg failed that test by aiming to “obliterate the distinction between possible existence and actual existence. If some universes are merely possible while others are real, what distinguishes the real ones? The theory I’ve outlined makes it unnecessary to ask such uncomfortable questions. Any universe that can exist does exist; there’s no longer any need to explain why ours was granted special privileges. They’re all real.”
The two worlds differ mainly in two ways. First, the real is causally efficacious; the ideal is inert and powerless on its own. Second, the real is finite; the ideal is infinite.
In the article on Landsburg linked to above, I argue that “it’s impossible to show that real things are infinite in number, because we’d literally have to count them one by one, and we’d never finish counting, staying at every moment at a finite number no matter how large.” So, there cannot be a proof that an actual infinite can exist.
One way to assure ourselves that it can’t is to consider that the ideal world, as regards abstract math, admits infinities of different sizes. Suppose then that there can be an actual infinite of, say, atoms in the universe. Is that infinity countable? Continuum? Still higher infinities that multiply without end? Why or why not?
The implausibility of infinite time stretching into the past, with the corresponding infinite number of events that would have occurred in the life of the universe up to now, is shown by William Lane Craig adequately in his Reasonable Faith. Another simple argument to that effect is that to every day in the past it should be possible to assign a date value, such as 1/1/10,000 BC. But -∞ is a mathematical abstraction and not a date. Hence it is unintelligible to postulate that time could have “began” an infinite number of days ago.
(Of course, the claim may be rephrased as that the universe never began, not even an infinitude of days ago. But I find that an even less credible idea.)
Note that the everlasting life promised to the saints is not a counterexample of my thesis, because it is a mere potential infinity. One lives forever, but at every moment remains to have lived only for a finite amount of time, to have had a finite number of experiences, etc.