### Comparing Infinities

This has been bugging me all day:

Comparisons involving infinitely large numbers are notoriously tricky. ... To grasp the mathematical challenge, imagine that you’re a contestant onLet’s Make a Dealand you’ve won an unusual prize: an infinite collection of envelopes, the first containing $1, the second $2, the third $3, and so on. As the crowd cheers, Monty chimes in to make you an offer. Either keep your prize as is, or elect to have him double the contents of each envelope. At first it seems obvious that you should take the deal. “Each envelope will contain more money than it previously did,” you think, “so this has to be the right move.” And if you had only a finite number of envelopes, it would be the right move. To exchange five envelopes containing $1, $2, $3, $4, and $5 for envelopes with $2, $4, $6, $8, and $10 makes unassailable sense. But after another moment’s thought, you start to waver, because you realize that the infinite case is less clear-cut. “If 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. ... 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... 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.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 7.1 Every whole number is paired with an even number, and vice versa, suggesting that the quantity of each is the same.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. ...

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.

Physicists call this the measure problem, a mathematical term whose meaning is well suggested by its name. ... Solving the measure problem is imperative.

[From Greene, Brian (2011).The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos(Kindle Locations 3609-3624). Random House, Inc.. Kindle Edition.]

Posted by Mark Thoma on Monday, December 12, 2011 at 12:24 AM in Economics, Science |
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