No two points on an infinite line are infinitely distant

I mentioned this in passing in "What if there was no beginning?" in connection with the Kalām Paradox. but I think it deserves a post of its own.

The Kalām Paradox, for those who came in late, asserts that nothing temporal can have always existed. If it had, it would be infinitely old, meaning that an infinite amount of time must already have elapsed. However, it is impossible for an infinite amount of time to elapse, because time elapses one finite step at a time, and adding up finite quantities can never yield an infinite quantity.

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First, I should make it explicit what sort of infinity we're talking about here. We're not talking about Zeno's paradoxes which find an uncountable infinity within countably finite quantities. In the racecourse paradox, for example, it is supposed to be impossible to run a mile, because first you have to run half a mile; and before you can run half a mile, you have to run a quarter of a mile; and so on -- an infinite number of tasks. A number-theoretical way of expressing this is that between any two integers there are (uncountably) infinitely many real numbers. To apply Zeno's spatial paradox to time, we could say it is impossible for anyone or anything to be a year old -- because before a year can elapse, 6 months have to elapse; before 6 months, 3 months; and so on -- a sum of infinitely many finite quantities, which must therefore add up to infinity. This is not the Kalām argument, which takes it for granted that things can have a finite age.

I will not bother to address Zeno-type paradoxes because no one is in any danger of believing them. The Johnsonian "I refute it thus!" is sufficient. It's very obvious that fleet-footed warriors can catch tortoises, and that it is in fact possible to run a mile. These are matters of everyday observation, as the subject matter of Kalām -- the question of whether the universe had a beginning -- cannot be.

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When the Kalām Paradox denies that the past can be infinite, it is referring to countable infinity, of which the paradigm case is not the real numbers but the integers. (To be sure, time may also be infinitely divisible and thus uncountably infinite, but that is a separate question.) The image we want to keep in mind, then, is the integer number line.

I propose that linear time is like this number line. The origin (zero) corresponds to the present, the negative integers to points in the past, and the positive integers to points in the future. Let's say that -1 on the number line means "one year ago," 1 means "one year hence," and so on. (There are of course many -- possibly infinitely many -- intermediate points in time between each of these points, just as there are infinitely many real numbers between one integer and the next, but that is not germane to Kalām. We take it for granted that Zeno is wrong and that countably-finite intervals of time can and do elapse.)

The integer number line is countably infinite -- meaning not that it is infinitely divisible but that it is infinitely long. There is no "first" or "last" integer. This does not mean that there is a first integer, -∞, which is infinitely distant from the origin; there is no such integer as -∞; there is no first (or last) integer. Every integer is finite (meaning finitely distant from the origin), and therefore the difference between any two integers is also finite.

Applied to time, this means that there was no first moment of time, no absolute beginning -- not that the beginning was "infinity years ago" (-∞), but that there was no beginning.

Now according to the Kalām Paradox, this means that an infinite amount of time must already have elapsed to get to the present. This is impossible, and yet we manifestly have reached the present; therefore, there was a beginning.

The question to ask is: An infinite amount of time must have elapsed from what point to the present? If the answer is "from the beginning," the argument fails, because the claim it is trying to disprove is that there was no beginning, just as there is no first integer. If the answer is "not from the beginning, since there is none, but from any arbitrary point infinitely distant from the present," the argument also fails, because there is no such point, just as there is no such thing as an infinite integer (infinitely distant from the origin). The only answer left, then, is "a point finitely distant from the present," and then there is no more paradox.

If there was no beginning, then we can say that, for any finite number n, n years have already elapsed. This is true no matter how arbitrarily big n is, which is the sense in which the past is infinite. But it does not mean that an infinite number of years have already elapsed. Therefore, this presupposition of the Kalām Paradox conclusively fails.

I am open to corrections from readers more mathematically gifted than myself, but I have to say I'm pretty darn sure I'm right about this.