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The Joy of Sets

By Jason Rosenhouse

In recent days, the pro-mathematics portion of the Internet has been buzzing over the following paragraph, taken from the website of Christian publishing company A Beka Book:

Unlike the “modern math” theorists, who believe that mathematics is a creation of man and thus arbitrary and relative, A Beka Book teaches that the laws of mathematics are a creation of God and thus absolute….A Beka Book provides attractive, legible, and workable traditional mathematics texts that are not burdened with modern theories such as set theory.

As a result of recent legislative activity in the state of Louisiana, these curricular materials will now be supported with taxpayer money.

In more than a decade of socializing with creationists and other religious fundamentalists, I frequently encountered blinkered arguments about mathematics. This attack on set theory, however, was new to me. I cannot even imagine why anyone would think set theory is relevant to discussions of whether it is man or God who creates math. Perhaps the problem is that set theorists often speak a bit casually about infinity, which some people think is tantamount to discussing God. Alas, this line of criticism is too blinkered to take seriously.

Whatever their objection, they are really missing out on something great. Set theory is fascinating.

By a “set” we mean simply any collection of objects. You walk into a grocery store and see a pile of grapefruits over here and a pile of apples over there. A mathematician might then refer to the set of grapefruits on the one hand and the set of apples on the other. This provides a useful way of talking about all the apples (or grapefruits) combined as one unit, as opposed to discussing any specific apple (or grapefruit).

Of course, we can identify many other sets. We might wish to distinguish the set of Gala apples from the set of Granny Smiths. Or we might want to make a larger set by combining the set of apples and the set of grapefruits together to form part of the set of all fruit. For any description you would care to give, it is reasonable to talk about the set of all things that fit that description.

This seemed obvious, for example, to Gottlob Frege, a German mathematician/philosopher who did pioneering work in logic and set theory in the late nineteenth and early twentieth centuries. Bertrand Russell pointed out that this notion is fundamentally flawed. He first observed that some sets answer to their own descriptions while others don’t. The set of all grapefruits isn’t itself a grapefruit. Therefore, this set doesn’t contain itself among its members. On the other hand, the set of all abstract ideas is, indeed, an abstract idea. So it contains itself.

Russell now considered the set whose members are precisely the sets that are not contained within themselves. The set of all grapefruits is contained in Russell’s set, for example, while the set of all abstract ideas isn’t. He now wondered whether his set did or didn’t answer to its own description. If we suppose that it does so answer, then it must be contained within itself. But Russell’s set only contains sets that aren’t contained within themselves. This is a contradiction. You see, if we assume that Russell’s set answers to its own description then it both contains itself and doesn’t contain itself. Impossible.

Alas, the alternative assumption fares no better. If we suppose that Russell’s set doesn’t answer to its own description, then it must be among the sets that aren’t contained within themselves. But this is precisely the criterion you must satisfy to get into Russell’s set in the first place. Either way you have a contradiction, meaning this isn’t a properly defined set.

Nor is this the only way to get into trouble with sets. Consider the set of counting numbers {1, 2, 3, 4, …} that cannot be uniquely identified with fewer than two hundred characters. For example, a number such as 1000 can be identified by writing “ten multiplied by one hundred,” but I can do it more efficiently by writing “one thousand,” and more efficiently still by writing “ten cubed.”

Now, since there are only finitely many phrases having fewer than 200 characters, and infinitely many counting numbers, it is clear that my set must contain something. And since it must contain something, it must also contain a smallest number. (In the math biz, this curious fact of counting numbers is known as the “well-ordering principle.”) That smallest number in the set is therefore uniquely identified by the phrase, “The smallest counting number that cannot be described with fewer than two hundred characters.” But did I not just describe it with fewer than 200 characters? Prolonged consideration of such things can be harmful to your mental health.

Actually, my favorite application of the well-ordering principle is this: Consider the set of all the boring counting numbers. This set must have a smallest member, let us call it X. But then X is the smallest boring counting number, which makes it very interesting indeed! Surely this contradiction shows that all counting numbers are interesting?

Indeed they are. And sets are as well. Just don’t be too ingenious about how you define them.

Jason Rosenhouse is Associate Professor of Mathematics at James Madison University. His most recent book is Among The Creationists: Dispatches from the Anti-Evolutionist Front Lines. He is also the author of Taking Sudoku Seriously: The Math Behind the World’s Most Popular Pencil Puzzle with Laura Taalman and The Monty Hall Problem: The Remarkable Story of Math’s Most Contentious Brain Teaser. Read Jason Rosenhouse’s previous blog articles.

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  1. [...] Oxford University Press features a vigorous defense of set theory that presents the ingenious examples that mathematicians and philosophers have come up with over the years. [...]

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