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If time were to be infinite in both the past and the future, does that mean that every event must occur an infinite number of times?

If time is only infinite in one direction, it is easy to see how this isn't necessarily the case. For example, a wave propagating. It can propagate infinitely in one direction, but still only be in one location once. It is also easy to assign a time to this event; t=0 at the "start" of this time, and t continues into infinity.

However, does this translate into time infinite in both directions? If we use the wave propagation example, the wave will never start propagating in a single direction (no start to an infinite past), and it seems that after an infinite time (without the future even existing yet), the wave will already have propagated fully into infinity. Secondly, does it make sense to assign a time to a single event? If an event only occurs once in this infinite time, in what sense can we say that this event is unique, in terms of the time at which it occurs?

Sorry if this is in the wrong site, it might be more of a physics question, as I'm wondering about whether infinite time in both directions, as implied by eternal inflation, means every event must be recurrent.

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    No, and it makes no difference whether time is infinite in one or both directions. Waves can propagate in one direction over the entire bi-infinite timeline, just take f(x-t) for any shape function f(x), not that it matters here. Recurrence is only implied (by the pigeonhole principle) when finitely many states must occur infinitely many times. And events are unique or not intrinsically, their time stamp is moot.
    – Conifold
    Jul 23, 2019 at 1:37
  • Both the Hindu and Buddhist texts say that time is infinite in both directions, and that events repeat themselves. If you roll a pair of dice many times, sooner or later the same pattern of rolls will repeat.
    – Swami Vishwananda
    Jul 23, 2019 at 10:09

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The issue is that it is not just the amount of time but the number of states that may take place within reality. If the number of states that may occur is finite then infinite time would suggest (but not demand) that each state occur infinitely. If the number of possible states is infinite then a particular state may occur once or not at all.

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On the question of repetition, no, we could have an infinite past and an infinite future without any repetition of past events.

There are different ways to think of this but they all come down to the same idea. The idea of repetition assumes that the space of possibilities is not infinite. As your metaphor suggests, we intuitively think in terms of a wave going through that space and since we think of it as being limited, we see the wave coming back and inevitably repeating itself.

However, there is no good reason to assume that the space of possibilities is limited. It could be infinite, too.

One way that the space of possibilities be infinite is that physical space itself be unlimited. This could make the space of possibilities infinite.

But our 3D-space could also curve like a 4D-sphere and therefore be limited in extension with nonetheless an infinite space of possibilities. For example, if there is no smallest physical distance (or, equivalently, that is, if there is always another point of physical space between any two different given points).

There are other possibilities.

For example, if at least one physical property has quantities along a continuum, without limits either way.

Another example would be if there was an infinity of distinct physical properties.

Well, etc.

So, no, no logical reason for repetition of past events. EB

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