Actual Infinity in Reality?

[Angrboda]

I provided my answer in another thread before I saw this thread. So I'll simply cut and paste that response here. Unless I've missed something, it does not appear that there are any logical contradictions to my conclusion here. If someone has one, please provide it. To sum up, it doesn't appear to be impossible for the universe to be temporally infinite. On less certain ground, I don't see anything immediately contradictory to the notion that the universe may be spatially infinite.

It is typically argued that the universe can't be past eternal because there would be no way to traverse an infinity of time. However, such arguments about being unable to traverse an infinite past up to the present, as implied by such arguments as the one that says you can't count successively to infinity, rely on the A theory of time. The idea of "traversing" an infinite past is incoherent on the B theory of time. If the B theory of time is correct, and the universe is infinite in time, such arguments do not apply and you have the case of an actual infinite existing. This leads to attempts to show that the universe is not past eternal by attempting to directly demonstrate that the universe's past is not infinite because the universe had a beginning. This is done by invoking things such as the Borde-Guth-Vilenkin theorem and the standard interpretation of the big bang model which supposedly indicates that the universe has a beginning. However there are theories such as Penrose's Conformal Cyclic Cosmology (see below) in which neither of these objections apply. Conformal Cyclic Cosmology explains why we would find points "in time" that have the appearance of resulting from a universe that had a beginning in a universe which does not in fact have a beginning. So, to the best I can tell, the idea that the universe is temporally infinite is consistent with a B theory of time and with some models of cosmology. So, ultimately, it doesn't appear that the case that you can't have an actual infinite has been made. (Regarding Hilbert's hotel, supposedly the results are absurd. This can mean several things. It can mean that the result is counter-intuitive, or it could mean that the result is logically impossible. I don't off-hand see that Hilbert's exercise demonstrates anything about logical impossibility so much as it is just showing that such things seem to defy our normal intuitions. I don't see the latter as any kind of argument that actual infinities don't exist so much as a demonstration that we aren't natively well equipped to think about such things. That latter fact is of little consequence. Quantum mechanics presents results that are equally absurd in that sense, that doesn't make quantum mechanics wrong. If you think Hilbert's hotel demonstrates something more substantial than this, I'd appreciate someone drawing out the relevant connections, because I don't see them.)

The conformal cyclic cosmology (CCC) is a cosmological model in the framework of general relativity, advanced by the theoretical physicists Roger Penrose and Vahe Gurzadyan. In CCC, the universe iterates through infinite cycles, with the future timelike infinity of each previous iteration being identified with the Big Bang singularity of the next. Penrose popularized this theory in his 2010 book Cycles of Time: An Extraordinary New View of the Universe.

Wikipedia || Conformal Cyclic Cosmology

The basic point is this. The very early universe is smooth. The universe right now is lumpy, with stars and galaxies and black holes all over the place. But the future universe will be smooth again — black holes will evaporate and the cosmological constant will disperse all the matter, leaving us nothing but empty space. (Just wait about 10100 years.) So, Penrose says, we can map the late universe onto a future phase that looks just like our early universe, simply by a conformal transformation (a change of scale). Do this an infinite number of times, and you have a cyclic cosmology — the universe goes through a series of “aeons” that start with a smooth Big Bang, get lumpy as structure forms, smooth out again, and then gets matched onto another smooth Big-Bang-like phase, etc.

Penrose’s Cyclic Cosmology