Given Another 100 Million years, What Are The Chances That I Will Exist Again?

It doesn’t matter, because the number of possible planets is much, much greater than the possible number of planets.

Suppose the universe is infinite in all directions, and it is chock full of earth-sized planets, packed as closely as gumballs in a jar.

You can still count them. Pick any one, then count the planets touching it. Then count the previously uncounted planets touching those. And so on. The total number will be infinite, but it will be countably infinite, like the number of integers, and surprisingly, like the number of rational numbers. In the Cantor transfinite series, that’s aleph-null.

Now consider a single leaf on a single tree on a single planet. It might be one inch long, or it might be two inches long, or it might be somewhere in between. Its exact length is a real number, and it has been proven that there are more real numbers between 1 and 2 than there are integers or rational numbers. Or, since they can be put into a one-to-one correspondence with the integers, the number of tightly packed planets that an infinite universe can hold.

In practice, the finite size of molecules means that there aren’t that many possible measurements for the length of that one leaf, but you also have to consider its width, weight, chemical composition, exact shape, etc. And if you want a planet identical to earth, you have to do that for every leaf on every tree, every limb on every animal, every rock and pebble and grain of sand, etc.

It’s not impossible that such a planet exists, but as I said in an earlier post, the chances of it are statistically zero.

The exact length of a leaf will not be “a real number” because it is made of atoms (and, if you’re going to nitpick, of subatomic particles). The length has to be one of a countable number of lengths, and there can only be a countable number of possible leaves. If you get down to a small enough length, the universe is apparently discrete, not continuous.