The Axiom of Choice

Can we choose an element from each set in a family of nonempty sets? Does every set have a well order? Is the axiom of choice arrogant overreach or does it rather express an enduring logical truth?

At the candy shop as a child, your fingers on the counter's edge, you peer up at the big jars of candy, filled with sweets. You place your quarter firmly on the counter, and the kind owner nods—you choose exactly one candy from each jar and are soon proudly carrying your treats out of the shop in a crisp white bag.

In mathematics, one often finds oneself in just such a situation, with a collection of nonempty sets and an urgent desire, for some mathematical purpose, to select a single item from each set in the family. Is it always possible? Can we always choose a mathematical sweet from each set? Perhaps it seems obvious—just pick an element arbitrarily, randomly choosing an item from each set. Can we expect to choose like the kid in the candy shop?

And yet, this principle of being able to choose an element from each set in a family of sets is sometimes viewed in mathematics as controversial. The objection is that in some cases we might not be able to describe any rule or constructive procedure by which we shall make the choices, and ultimately the worry is that perhaps in those kinds of cases there may actually somehow be no coherent way of choosing. The perspective is that mathematical existence claims should be deeply connected with the constructions by which we may realize them. And so some mathematicians therefore hesitate to agree that the principle holds universally, and many keep careful track of exactly which mathematical arguments make use of the choice principle and which do not. Meanwhile, other mathematicians find the principle unproblematic and make a free use of it—they take the axiom of choice as asserting a fundamental feature concerning the nature of sets, a logical truth about which kinds of sets there are.

Enjoy this new installment from The Book of Infinity, a series of vignettes on infinity with all my favorite puzzles and paradoxes, serialized over the past year.

This edition features the axiom of choice, including a clear elementary proof of the well-order theorem, Zorn’s lemma, the Vitali construction, the Banach-Tarski paradox, the intriguing possibility of uniformly definable global choice functions, a discussion of extrinsic and intrinsic justification in the foundations of mathematics, and much more. For a special treat, an elementary account is given of the self-replenishing gumball machine, a simple toy version of the Banach-Tarski result.