A Paradox of Time

Suppose that time is cyclical. It begins somewhere, and never ends. It follows that there is some finite n which is the number of the cycle that we are currently in. Let the first cycle be numbered one. Supposing that we have no knowledge about what cycle we are in, what do we estimate n to be? We can state the question differently: what is the average value of n?

There are infinite possible values for n. So for every n, P(n) = 1/infinity. So the average value A of n is A = (1 * 1/infinity) + (2 * 1/infinity) + … = lim as x->infinity of (1/infinity) * (x+1) * (x/2) = 0. So on average, we are in cycle zero. The first cycle, however, is numbered one; so this is not a cycle at all. It appears that on average we are outside time, before the beginning of the universe.

Of course, if we postulate that time is cyclical, but has neither a beginning nor an end, then on average — and in fact, at all times — we are on cycle infinity. This is a different way of stating that at every time, there are infinite number of cycles before us, and an infinite number of cycles ahead of us.

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