Do the Gods repeat themselves?

This is a question that's been floating around in cosmological circles.

For example:

If space goes on forever, then there must be other regions like ours — in fact, an infinite number of them. No matter how unlikely it is to have another planet just like Earth, we know that in an infinite universe it is bound to happen again.

Your hear that a lot from cosmologists, but it's shoddy thinking. An infinite space doesn't have to contain every possible combination of matter. It might just contain an infinity of Cabbage Patch Dolls arranged at the intersections of a cubic lattice. Or it might, beyond a certain point, just be empty. The positive odd integers 1, 3, 5, 7, 9, 11, … are an infinite set, but the set doesn't contain any even integers.

There are serious problems with this argument. First of all, we have a problem with the definition of "possible". While it's true that the set of positive odd integers is infinite, and while it's true no even integers appear in it, it also reeks of – well, design. If you pulled numbered balls out of a barrel, and all you ever found was even numbers, you might be tempted to conclude that the odd numbers had been excluded. By defining a set as "the positive odd integers", Derbyshire has defined a set which specifically excludes even numbers. Thus, under this definition, it's impossible for an odd integer to appear in the set.

Scientists work with a number of basic assumptions – axioms, if you will.

One of these is, if something exists, it's possible.

Since the Earth exists, it is, by definition, possible. (It happened.)

Since it's possible, the probability that it would actually happen in some volume of space is greater than zero. (In contrast to the probability of finding an even integer in the set of positive odd integers.)

Science also assumes no one or nothing is stacking the deck. There is no reason to assume that, because anything has appeared once, its chance of appearing in the next region of space is changed in any way. (That is, the "gambler's fallacy" doesn't apply to natural events.)

In particular, if the earth can arise in the universe once, it must have some chance of appearing in any other volume of space, however low that chance may be. And if you examine a large enough volume of space, you will hit the jackpot sooner or later.

The trick is to live long enough.

If the odds are sufficiently long, there may be no other Earths close enough to ever show up in our radar.