Many of us have a sense for physical units like 750 mL, 12 ounces, or a six-pack. But what does it mean to have an infinite size?

The math is fascinating. Perhaps the most remarkable result is that infinity is not a single size. One can talk intelligently about different sizes or types of infinity. This was first and famously demonstrated by Georg Cantor in the late 1800s, in work that laid groundwork for the field of set theory.

Set theory is one of my favorite subjects, but perhaps it is not for everyone. Luckily there is another way to approach infinity because of a new proof using game theory.

What is the size of a set of objects?

Size is the most basic property of a set. It is the number of objects in a set. We refer to the size of a set when we say things like “there are 5 people in front of me” or “a theater has 200 seats.”

Sometimes we do not care about absolute size but instead comparative size. For instance, in an election, it is most important to know which candidate gets more votes. What we usually do is count the number of votes for each side and then make a comparison.

But amazingly we don’t have to count to compare the size of two things. My favorite set theory textbook has the explanation of why:

To see how this is done, consider the problem of determining whether the set of all patrons of some theater performance has the same number of elements as the set of all seats. To find the answer, the ushers need not count the patrons or the seats. It is enough if they check that each patron sits in one, and only one seat, and each set is occupied by one, and only one, theatergoer.

In essence, comparing size is equivalent to the process of pairing elements. If elements from two sets pair up exactly one-for-one, then the sets are the same size. If one set has unpaired elements (say unfilled seats), then it is bigger.

This is common sense when dealing with finite sets, but things get nuanced and defy physical logic when we analyze infinite sets.

Countably infinite

The numbers we count with (the set of numbers {1, 2, 3, …}) are called natural numbers. The set keeps on going forever and so there are an infinite number of elements. Because we can “count” how many objects are in the set, the set’s size is described as being countably infinite.

How do other sets compare in size? The surprising part is that many sets are also countably infinite, even though they would first appear to be smaller or larger. We have to use the pairing process and not our intuition to decide the size.

For instance, consider the set of numbers {2, 3, 4, …}, that is, the natural numbers except for 1. Though this set has one element missing, it is in fact the same size as the natural numbers. The reason is it can be paired up exactly one-for-one with the natural numbers. Every number n from the natural numbers gets paired with the number n + 1 from this new set. Since both sets keep going on indefinitely, the pairing is proper and works. Hence the new set, though seemingly smaller, is also countably infinite.

The result can be hard to swallow in a physical sense. Imagine an infinite theater with seats labeled with the natural numbers that is originally full with people. The set {2, 3, 4, …} can be thought of physically as the seating arrangement of people if we asked the first person to leave the theater and then asked everyone to move back one seat. The theater would somehow still be full despite losing one person. And even more strange is that if the first person returns, we could clear the first seat by asking everyone to move up one seat.

One mathematician, Hilbert, presented many other similar paradoxes of the infinite. He used an infinite hotel and so such problems fall under Hilbert’s Hotel paradox.

The physical analogies make the math seem absurd, but in fact everything is sound and well-derived. So I keep myself straight by thinking about things as abstract pairings.

Other sets are also countably infinite, even though they too seem to be of different size. The even numbers are exactly the same size as the natural numbers, as each number n can be paired with 2n. Similarly, the odd numbers are countably infinite, and so is any infinite subset of the natural numbers, like the prime numbers.

Amazingly the set of rational numbers–the fractions of natural numbers (and their negatives)–is also countably infinite. Here is one illustration of how you can “count” the fractions.

At this point every set we have come up with is countably infinite. So it’s worth asking: are there any sets bigger? It turns out there are bigger sets that are “uncountably” infinite. This is where the game theory proof comes in.

Game theory: proof of the uncountably infinite

I’ll summarize the proof below.

The game involves two-players, Alice and Bob, picking numbers on the interval [0,1]—the set of real numbers (all the decimals) between the numbers 0 and 1. Here are the rules:

1. Alice initially picks some subset S of the interval [0,1].
2. Alice starts the game by picking a number a₁ between 0 and 1.
3. Bob then picks a number b₁ bigger than Alice’s choice a₁ but less than 1.
4. Alice then picks a number a₂ bigger than her previous choice a₁ but less than Bob’s previous choice b₁.
5. In general, Alice and Bob alternate picking numbers between the two previous numbers played. That is, on turn n Alice has to pick a number a_n such that a_n-1 < a_n < b_n-1 and Bob has to pick a number b_n such that a_n < b_n < b_n-1.
6. A famous theorem guarantees that the sequence of numbers a₁, a₂, … converges to a number. Call this number X.
7. If X is in the set S, then Alice wins. Otherwise Bob wins.

The proof that [0,1] is uncountably infinite

The above proof has three parts:

1. If S is countably infinite, then Bob has a winning strategy.
2. Alice can always win the game.
3. The set [0,1] is uncountable.

Here are the details on the individual steps.

1. If S is countably infinite, then Bob has a winning strategy.

Imagine the set S is countably infinite. Then the set can be enumerated as s₁, s₂, and so on (like seats in an infinite theater). It turns out Bob can always make sure that the sequence does not converge to any of these values. That is, Bob can make sure X does not equal any s_n.

Here’s the strategy: on turn n, Bob picks s_n if it is a legal move, and otherwise picks anything. This will guarantee it.

Why? If Bob does get to pick s_n, then he forces the remaining numbers to be smaller than s_n so the sequence cannot converge to s_n.

On the other hand, if Bob cannot pick s_n, then he can pick any legal move. The only reason Bob could not pick s_n is if Alice has already picked a number higher than s_n. Accordingly, the remaining numbers picked in the game will be larger than s_n so the sequence would converge a number larger than s_n. Thus Bob can just pick any allowable number on this turn and continue.

Bob can continue this strategy to make sure that the sequence never converges to any element of S. This means X would not be in S and Bob would win. In summary, if S is countably infinite, Bob can always win the game.

Unfortunately, the odds are stacked against Bob.

2. Alice can always win the game.

Alice can pick S to be any subset of the interval [0,1]. This means Alice can pick the entire subset [0,1]–cheap, yes, but effective. Since Alice and Bob are always picking numbers between 0 and 1, the sequence must converge in the interval. Hence Alice can always win the game.

3. The set [0,1] is uncountably infinite.

This follows directly! By point 1, if [0,1] were countably infinite, then Bob could win the game. But by point 2 he cannot win, and therefore the set [0,1] must be larger and uncountably infinite.