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Time stretches back to infinity, therefore it cannot have reached NOW {let 2009 = NOW}. Manifestly, however, it has reached NOW. How can this be?

As a warm-up exercise, consider the following two infinite ordered sets of numbers. Firstly, take the negative and positive integers in their 'natural' ordering ... -4, -3, -2, -1, 0, 1, 2, 3, 4, ... trailing off unendingly to the left and to the right. Second, take all the negative numbers, in increasing size, followed by zero and all the positive numbers: -1, -2, -3, -4, ... o, 1, 2, 3, ... Now, in both orderings, any positive number is preceded by an infinity of numbers (including all the negative numbers). But there is a very important difference between the two cases -- they have, as mathematicians say, different 'order types'. One big difference is this: there is no first member of the first ordering (i.e. for any given element of the ordered series, there's an earlier one); but there is a first member of the second ordering (namely, -1). To bring out another difference, suppose in each ordering we take one of the negative numbers, and we ask: can we...

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