I think it's sophistry to pretend we haven't any more idea of that since pre-Democritus. Thousands of years of science has shown that infinities are always a problem in our heads, with our theories. Does that prove they don't exist? No more than it's proven that the sun will come up tomorrow, I guess.
Just to be clear why this whole conversation thread is a TypeError, let's say I assign a probability of 99.9% to the hypothesis that the state space of the universe is finite, and 0.1% to the state space of the universe being infinite...
... In that case, how big is my hypothesis space about possible states of the universe?
There are plenty of examples of inifinities that are not problematic. Infinitely small wavelengths make our current understanding of physics break down, indeed. Or maybe infinitely divisible solids that lead to paradoxes like Banach-Tarski's. On the other hand, infinitely dimensional configuration spaces or continuous parameterization (e.g. coordinates, field strengths, phases) are trivial unoffensive parts of classical and quantum mechanics.
> Do you agree with Aquinas’ corollary, that absent an actual infinity, there must be some First Cause, which we call God?
Note that even if you agree with Aristotle’s position, which is essentially an arbitrary assumption, and the corollary that there must then be a first cause, there's nothing except the boat of being stepped in a particular religious tradition to suggest that the first cause should have any of the other traits of any particular concept of God. It works just as well to take the earliest known thing on the sequence of causes and say “this cause is uncaused”.
Aquinas' argument is that there can't be an actual infinity, so even though it appears that everything has a prior cause, it must be that there is something which is self-causing. "The Big Bang" qua event clearly didn't cause itself (events only cause events that are later in time), so the typical way to cash this out is "the Big Bang" qua set-of-physical-laws is self-causing.
This leads to a new problems (why this set of laws vs. some other), unless you posit that the laws are somehow perfect or necessary (which is essentially Deism), but the laws of our universe seem to be contingent (lots of unexplained physical constants).
You can make a metaphysically plausible case of a Big Bang-Big Crunch cycle that goes on forever, but then you're back to believing in an actual infinity.
> This leads to a new problems (why this set of laws vs. some other), unless you posit that the laws are somehow perfect or necessary
Those problems are only problems with the aesthetic preference that the universe be perfect or necessary. Once you accept that the universe can be without adhering to any such aesthetic preference, they cease to be problems.
So then the universe existing is a brute fact with no cause (as opposed to being self-causing).
You can do that but once you say there are facts without causes it's hard to know what you're signing up for. Why is this the brute fact and not something else? By definition, there is no answer (no cause) for that question. Uh okay, but if brute facts are possible, how can we do science at all? For all we know, we're just surrounded by brute facts and attempts to systemize facts into theories is just a waste of time because a new brute fact can just come along and bite you in the ass tomorrow. But I thought we only posited brute facts because science was pushing us in that direction by show us that there was a Big Bang, but now suddenly we're told "science is only contingently possible and sometimes just fails entirely due to the existence of brute facts".
It's not a satisfying intellectual stance, and if you really poke at it, it just feels like motivated reasoning in which the conclusion (there is no God) is leading the premises (some facts have no causes), not the other way around.