The Axiom of Choice: How to Pick One Thing from Each of Infinite Bins

Here’s a puzzle. Imagine you have a hundred bins, each containing some objects. If you want to pick one object from each bin, you can just walk down the line and grab something. No problem. Even if you had a thousand bins, or a million, you could still do it — it would just take a really long time.

But what if you have an infinite number of bins? You can’t walk down an infinite line. You can’t say “keep going until you’re done” because you’d never finish. And yet, mathematically, it sure seems like there should be a way to pick one thing from each bin. The bins exist. The things inside them exist. Why wouldn’t there be a way to select one from each?

This is the puzzle at the heart of the Axiom of Choice. It sounds obvious, and many mathematicians think it must be true. But it leads to some of the strangest, most mind-bending results in all of mathematics — including a proof that you can cut a solid sphere into pieces and reassemble them into two spheres the same size as the original. And the whole thing has been the subject of huge debates for over a hundred years.

What the Axiom Actually Says

The Axiom of Choice (let’s call it AC for short) was first stated clearly by a mathematician named Ernst Zermelo in 1904. Here’s what it claims:

If you have a collection of non-empty sets, there exists a function that picks exactly one element from each set.

That function is called a “choice function.” So AC says: for any collection of bins with things in them, you can simultaneously grab one thing from each bin — even if the collection is infinite.

For a simple example, imagine you have the set of all non-empty subsets of the numbers 0 and 1. That’s just three sets: {0}, {1}, and {0,1}. You can easily define a choice function: pick 0 from {0}, 1 from {1}, and 0 from {0,1}. That’s one choice function. You could also pick 1 from {0,1} instead. No problem — there are two.

But now imagine something much harder. Suppose you have an infinite collection of pairs of real numbers. For example, the pair {0, 1}, {2, 3}, {4, 5}, and so on forever. You could try to define a choice function by picking the smaller number from each pair. That works, because every pair of real numbers has a smaller one.

But what if the pairs aren’t ordered? What if they’re just arbitrary pairs of real numbers with no rule telling you which is smaller — and the pairs are infinite in number? AC says there still exists a choice function. It just doesn’t tell you how to define it or describe it.

That’s what made people uncomfortable.

The Controversy

When Zermelo first proposed AC, many mathematicians hated it. They said: you can’t just claim something exists without showing how to build it or define it. That’s not mathematics — that’s magic.

The French mathematicians Baire, Borel, and Lebesgue — three of the biggest names in the field at the time — led the attack. They were what we’d now call “constructivists”: they believed that a mathematical object only exists if you can actually construct it. Saying “there exists a choice function” without being able to describe it seemed like cheating.

Zermelo defended himself. He pointed out that AC could be stated in a way that didn’t even mention “choices” at all. Here’s his reformulation:

If you have a collection of non-empty sets that don’t overlap with each other (they’re “disjoint”), there exists a set that contains exactly one element from each of them.

Think of it this way: suppose you have a bunch of different-colored bins scattered around a warehouse, no two overlapping. There are objects in each bin. AC says there’s some collection of objects you can gather that contains exactly one from each bin. That “transversal,” as it’s called, just is a choice — but you can think of it as a set sitting there, without needing to imagine the act of choosing.

But this didn’t settle the debate. The question remained: do these choice functions and transversals actually exist in the mathematical universe, or are we just pretending they do?

The Banach-Tarski “Paradox”

The most shocking consequence of AC came in 1924, when mathematicians Stefan Banach and Alfred Tarski proved something that seemed to break the laws of physics.

Using AC, they showed you can take a solid sphere, cut it into a finite number of pieces (some of them extremely weird shapes — not like orange slices), and then reassemble those pieces into two spheres, each exactly the same size as the original. You could even reassemble them into a sphere the size of a house.

This is called the Banach-Tarski paradox. It’s not a contradiction in logic — it’s just extremely strange. It seems to violate common sense ideas about volume and conservation of matter. The trick is that the pieces are so weird and scattered that they don’t have a well-defined volume in the normal sense. AC allows these weird pieces to exist.

For many mathematicians, the Banach-Tarski paradox was a reason to reject AC — how could a true axiom produce something so bizarre? For others, it just showed that our intuitions about volume and infinity are limited, and that mathematics is stranger than we think.

Can We Prove or Disprove the Axiom?

For decades, mathematicians tried to figure out whether AC was actually true — whether it could be proved from the other axioms of set theory, or whether it contradicted them.

In the late 1930s, Kurt Gödel proved that AC is consistent with the other axioms of set theory. That means you can accept AC without introducing a contradiction. (Gödel built a special mathematical universe — the “constructible universe” — where everything has a definition, and he showed that in this universe, AC holds.)

Then in 1963, Paul Cohen proved the opposite: AC is independent of the other axioms. That means you can also reject AC without introducing a contradiction. Both choices are mathematically possible.

So AC is like a free-floating switch. You can flip it on or off, and either way you get a perfectly consistent mathematical world. The question isn’t “Is AC true?” but rather “Which world is more interesting or useful?”

Zorn’s Lemma: A Different Way to Say the Same Thing

There’s a principle called Zorn’s Lemma that’s equivalent to AC — if you accept one, you have to accept the other. Zorn’s Lemma is a bit technical, but here’s the idea:

Imagine you have a collection of objects where some are “bigger” than others (like building blocks where one block can contain others). Zorn’s Lemma says: if every chain of objects — every sequence where each one is bigger than the last — has some object that’s at least as big as all of them, then there must be a maximal object somewhere — one that nothing is bigger than.

This sounds abstract, but it’s incredibly useful. Mathematicians use Zorn’s Lemma all the time to prove that certain things exist — like a basis for every vector space, or a maximal ideal in a ring. These are things that AC guarantees exist, but you couldn’t build by hand.

Does the Axiom of Choice Destroy Logic?

Here’s a really strange twist. In the 1970s, a mathematician named Radu Diaconescu discovered that AC, when combined with a certain kind of logic, actually forces you to accept the Law of Excluded Middle — the principle that every statement is either true or false.

This matters because there’s a whole branch of mathematics called intuitionistic logic (or constructive logic) that rejects the Law of Excluded Middle. Intuitionists say you can’t call something “true” unless you can prove it. They don’t accept “Either the millionth digit of pi is 7, or it isn’t” as a meaningful statement unless you can figure out which.

Diaconescu showed that if you accept AC in an intuitionistic setting, you accidentally accept the Law of Excluded Middle too — which means you’re no longer doing constructive mathematics. This surprised a lot of people.

The resolution is that the kind of “choice” constructive mathematicians accept is different from the full AC. In constructive mathematics, saying “for every x there exists a y” already means you have a method for finding y — so the choice function is built in. But the full AC says something stronger: it says the choice function exists even when you have no method or description at all.

So What’s the Verdict?

Nobody has “settled” the Axiom of Choice. It’s not the kind of thing that gets settled. Most mathematicians today accept it because it’s incredibly useful and makes proofs much easier. A smaller number reject it, or try to work without it, because they find its consequences too strange.

But the real lesson is bigger than AC itself. The Axiom of Choice shows us that even the most obvious-seeming mathematical statements — “you can pick one thing from each bin” — can turn out to be deep, controversial, and connected to the foundations of logic itself. Mathematics isn’t just about calculating answers. It’s about deciding what we’re willing to accept as true, and seeing where that decision leads.

Appendices

Key Terms

Term	What it does in this debate
Axiom of Choice (AC)	Claims that for any collection of non-empty sets, you can pick one element from each — even if the collection is infinite and you have no rule for picking
Choice function	A function that picks exactly one element from each set in a collection
Transversal	A set that contains exactly one element from each set in a collection of disjoint (non-overlapping) sets
Zorn’s Lemma	A principle equivalent to AC that says certain collections of objects must contain a “maximal” one — useful for proving things exist
Banach-Tarski paradox	A surprising proof (using AC) that a solid sphere can be cut into pieces and reassembled into two spheres the same size
Law of Excluded Middle	The logical principle that every statement is either true or false — rejected by some mathematicians
Constructive mathematics	A branch of mathematics that only accepts objects you can actually build or describe
Intuitionistic logic	A system of logic that doesn’t assume the Law of Excluded Middle

Key People

Ernst Zermelo — German mathematician who first stated the Axiom of Choice in 1904 and used it to prove that every set can be well-ordered, sparking a huge debate
Kurt Gödel — Mathematician who proved in the 1930s that AC is consistent with other set theory axioms (you can accept it without contradiction)
Paul Cohen — Mathematician who proved in 1963 that AC is independent of the other axioms (you can also reject it without contradiction)
Stefan Banach and Alfred Tarski — Mathematicians who used AC to prove the “paradoxical decomposition of the sphere” (cutting one sphere into two)
Radu Diaconescu — Mathematician who discovered that AC implies the Law of Excluded Middle in certain logical systems

Things to Think About

If you had an infinite collection of bins and could somehow make infinite choices instantly, would you say the “choice function” exists even if nobody could describe it? Or does existence require some kind of description?
The Banach-Tarski paradox seems to violate common sense — but does it really violate logic? What’s the difference between something being impossible in the real world and something being a contradiction in mathematics?
If the Axiom of Choice is neither provable nor disprovable from the other axioms, does it make sense to ask whether it’s “true”? Or is “true” the wrong word here?
Imagine you’re a mathematician who only accepts things you can construct. You reject AC. What do you lose? (For example: you might not be able to prove every vector space has a basis.) What do you gain?

Where This Shows Up

Voting and elections — The idea of a “choice function” shows up in social choice theory, which studies how to combine individual preferences into a group decision (think: ranked-choice voting)
Economics — Economists use AC-related principles to prove that certain “ideal” economic outcomes exist, even when they can’t be practically computed
Computer science — The debate between constructive and non-constructive mathematics connects to questions about what computers can and can’t compute
Everyday infinity — Whenever you say “you could always just pick one” without specifying how, you’re relying on the spirit of AC — and it might not be as simple as it sounds