Eternal Recurrence
This follows up on my last post about the infinitude of time. If time is infinite, then probably there is eternal recurrence. Meaning: every qualitative occurrence will happen again, to any desired (but not 100%) degree of accuracy, infinitely many times. Given the infinite past, it also has happened infinitely many times before.
So there will be a person looking like you, sitting in front of a computer (or phone, or whatever) like the one you're looking at, reading a post containing these words, some time in the distant future. Also, this has all happened before, infinitely many times (again, assuming there's no beginning of time).
Historical side notes: This amazing view was first embraced (that I know of) by Seneca. Nietzsche famously entertained it as a thought experiment in The Gay Science. In later writings, it emerges that he thought that eternal recurrence was guaranteed by physics. Probably the philosophers who held this view, however, had an oversimplified view. They may have mistakenly thought that there were only finitely many possible states for the world, or only a finite range of possible states. They probably thought that the world would repeat its state exactly, rather than (as I claim) only to any desired degree of approximation. Also, Nietzsche appears to have assumed that the future occurrences of people looking like you would in fact be literally you. Obviously, that would be a controversial assumption that he doesn't argue for.
Anyway, we get the eternal recurrence from the Poincare Recurrence Theorem. To apply the theorem, you have to assume, basically, that the universe stays within a finite region. (If it just expands forever, with no upper size bound, then obviously it will never repeat its state, not even approximately.) Given that assumption, an infinite future gives you eternal recurrence.
Here is why the infinitude of the past is also interesting. If the universe has always existed, then it cannot be headed in any qualitative direction overall. If it was overall headed toward some state, then it would have long since reached that state, given an infinite past. For example, if the universe was overall headed toward increasing entropy, then it would have reached thermal equilibrium (heat death), an infinite time ago. If the universe was overall getting bigger, then it would have long since exceeded any assigned size, which means it couldn't be its current size.
If the universe had a beginning in time, then it started out in an incredibly low entropy state; also started out incredibly small. In that case, it might be that it expands infinitely, without bound. But if it has an infinite past, then it must be that it goes through cycles. Right now, it's expanding and entropy is increasing, but later (and before) it will be (and was) shrinking and entropy was decreasing. So the infinite past helps make the case for eternal recurrence.
I don't know whether the eternal recurrence is good news or bad news. At least it's more interesting than the scenario where everything expands and decays until it all ends up as photons spreading out through otherwise empty space forever.