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By Root 2084 0

I take the deal,” you think, “I’ll wind up with envelopes containing $2, $4, $6, and so on, running through all the even numbers. But as things currently stand, my envelopes run through all whole numbers, the evens as well as the odds. So it seems that by taking the deal I’ll be removing the odd dollar amounts from my total tally. That doesn’t sound like a smart thing to do.” Your head starts to spin. Compared envelope by envelope, the deal looks good. Compared collection to collection, the deal looks bad.

Your dilemma illustrates the kind of mathematical pitfall that makes it so hard to compare infinite collections. The crowd is growing antsy, you have to make a decision, but your assessment of the deal depends on the way you compare the two outcomes.

A similar ambiguity afflicts comparisons of a yet more basic characteristic of such collections: the number of members each contains. The Let’s Make a Deal example illustrates this, too. Which are more plentiful, whole numbers or even numbers? Most people would say whole numbers, since only half of the whole numbers are even. But your experience with Monty gives you sharper insight. Imagine that you take Monty’s deal and wind up with all even dollar amounts. In doing so, you wouldn’t return any envelopes nor would you require any new ones, since Monty would simply double the amount of money in each. You conclude, therefore, that the number of envelopes required to accommodate all whole numbers is the same as the number of envelopes required to accommodate all even numbers—which suggests that the populations of each category are equal (Table 7.1). And that’s weird. By one method of comparison—considering the even numbers as a subset of the whole numbers—you conclude that there are more whole numbers. By a different method of comparison—considering how many envelopes are needed to contain the members of each group—you conclude that the set of whole numbers and the set of even numbers have equal populations.

Table 7.1 Every whole number is paired with an even number, and vice versa, suggesting that the quantity of each is the same.

You can even convince yourself that there are more even numbers than there are whole numbers. Imagine that Monty offered to quadruple the money in each of the envelopes you initially had, so there would be $4 in the first, $8 in the second, $12 in the third, and so on. Since, again, the number of envelopes involved in the deal stays the same, this suggests that the quantity of whole numbers, where the deal began, is equal to that of numbers divisible by four (Table 7.2), where the deal wound up. But such a pairing, marrying off each whole number to a number that’s divisible by 4, leaves an infinite set of even bachelors—the numbers 2, 6, 10, and so on—and thus seems to imply that the evens are more plentiful than the wholes.

Table 7.2 Every whole number is paired with every other even number, leaving an infinite set of even bachelors, suggesting that there are more evens than wholes.

From one perspective, the population of even numbers is less than that of whole numbers. From another, the populations are equal. From another still, the population of even numbers is greater than that of the whole numbers. And it’s not that one conclusion is right and the others wrong. There simply is no absolute answer to the question of which of these kinds of infinite collections are larger. The result you find depends on the manner in which you do the comparison.10

That raises a puzzle for multiverse theories. How do we determine whether galaxies and life are more abundant in one or another type of universe when the number of universes involved is infinite? The very same ambiguity we’ve just encountered will afflict us just as severely, unless physics picks out a precise basis on which to make the comparisons. Theorists have put forward proposals, various analogs of the pairings given in the tables, that emerge from one or another physical consideration—but a definitive procedure has yet to be derived and agreed upon. And, just as in the case of