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If we divide time into individual frames, then we would get a set of infinite frames. But what is the cardinality of such a set?

Since time is continuous, like the real numbers, I would expect the cardinality of all frames in time to have the same cardinality as the real numbers.

However, my friend disagrees, and I'm not sure if my logic is good enough. So, would the cardinality of the set of all frames in time be the same as the cardinality of real numbers?

user2108462
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    I've thought about this before, and tend to agree with you, that time is like the continuum. We don't really know the cardinality of the real numbers though, we denote it as $\mathfrak{c}$ for continuum. However, it is consistent that the cardinality of the real numbers is $\aleph_1$, the first uncountable cardinal. – Rustyn Sep 26 '14 at 20:55
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    There is no way to tell (yet?). Perhaps time is actually discrete, but it just feels continuous since its granularity is so fine. – Mike Earnest Sep 26 '14 at 21:08

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If you begin by stipulating that the time line is a subset of the real numbers, then either the frames themselves are densely ordered (there is no notion of "next frame" available) in which case any infinite cardinality up to that of the continuum is an option (from $\Bbb Q$ to $\Bbb R$).

But I expect that if you use the term "frame" then you can at least move from one frame to another, in which case the time line is a well-ordered (or at least a scattered order) and therefore you can only keep track of countably many frames.

In general the question itself is very vague to give an exact answer, and will depend on too many variables that you haven't addressed: what is the original assumption on the time line, are the frames discrete, do the frames have some positive length (e.g. Planck time is a positive length)? The answer to these questions might change the answer to how many frames can be at all.

Asaf Karagila
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  • Ah, yes, whether if you can truly have a "next frame" was the question that was puzzling me. If time is continuous, then surely it is densely ordered? – user2108462 Sep 28 '14 at 13:46
  • You need to better define what continuous means here. Note by the way, that it might be the case that you prefer a different underlying order as the superstructure of the time line. – Asaf Karagila Sep 28 '14 at 15:02