The myth of eternal recurrence

(A thought that occurred while reading Schopenhauer)

The ancient idea of the eternal repetition of events has always struck me as very strange, though apparently Schopenhauer and Nietzsche, for example, didn't think so. (Supposedly, Eternal Recurrence is the normal world view of cultures before Judaism and Christianity, which are said to have originated the idea of history. Though I think C.S. Lewis pointed out that the Scandinavian pagans also had an idea of history—one involving a world-ending catastrophe to which one simply had to submit heroically . . . something like what happened in the final episode of the Angel spinoff of Buffy the Vampire Slayer . . . )

The mistake of these pagans, it occurred to me, is to think of the infinity of time as being larger than the infinity of all possible events. Modern mathematics does in fact acknowledge infinities of different sizes: for example, the number of irrational numbers is known to be larger than the number of rational numbers (fractions), although both are infinite. But intuitively at least, the infinity of possible time seems much smaller than the infinity of possible events. (Even if there were just a single spatial dimension, instead of three, there would appear to be an astronomical number of permutations possible for configurations of events on that dimension; far too many to fit on a single time dimension.) It follows from this, apparently, that one cannot fit all possible events in infinite space even once, let alone with an infinite number of repetitions, into an infinite time-line. One will never be able to duplicate the present state of the universe by looking at some point in the distant past or future. One would only be able to find patterns that were broadly similar to the present state. Any cycles in history would be approximate, not perfect, repetitions.

(A modernized version of the Myth might instead make use of the "many-worlds interpretation" of quantum mechanics. An unimaginably large number of different versions of the universe are said to exist side-by-side, in "Hilbert space" if I remember my physics correctly, as a result of the "splitting" of the universe into different possible outcomes that occurs with each probabilistic quantum event. Perhaps some of those alternate universes are identical to this one.)