the first post in a finite series
If there’s one thing about math that people love—and to make it through the average day, I have to believe there’s at least one—it’s infinity.
Throw the word into a math lesson, and ears perk up. Infinity? Did he say infinity? It’s like a distant celebrity, the subject of endless gossip and rumor. “I heard infinity isn’t even a number!” “Only the universe is really infinite.” “My last teacher said infinity times two is the same as infinity.” “I can use infinity to prove that 1 = 0!”
Infinity is a sound too high for our ears, a light too bright for our eyes, a taste so sweet that it would tear through our tongues like acid. Basically, it’s mathematical Mountain Dew.
Tellingly, all of our words for infinity define it by what it isn’t. Infinite: not finite. Unlimited: not limited. Boundless: without bounds. It’s hard to articulate what infinity does, so we settle for naming what it doesn’t: end. Infinity is the Anansi of mathematics, a trickster spider weaving baffling webs of paradox and contradiction.
Take this example: which has more numbers, List A or List B?
Well, they’re both continue forever, so they’re both infinite. But the first list includes every number on the second (2, 4, 6, 8…), and lots more as well (1, 3, 5, 7…). Thus, clearly, List A is the bigger group of numbers. Right?
Wrong. The lists are exactly the same size.
To see why, just go through list A and double every number on it, like so:
Now, doubling each number shouldn’t affect the size of the list. We haven’t added any items, or taken any way. It’s like taking a group of people and giving them fake beards—you don’t create or destroy any humans in the process.
And yet… without changing its size, we’ve somehow turned list A into list B. That can only mean that the two lists were the same size to begin with.
What’s happening here? Throughout our lives, we’ve relied on a simple rule for comparing the sizes of groups: If one group has all the members of another, plus some others, then it must be bigger. For example, if your list of Facebook friends includes all of mine, plus you’re friends with Kanye West, then you have more friends than I do (and probably cooler weekend plans, too).
And that’s perfectly true, for finite groups. But infinite groups… those are another matter.
Infinity shreds our intuitions. The startling truth is that all infinite lists are the same size, regardless of whether one happens to include the other.
The mathematician David Hilbert captured this paradoxical truth with the parable of Hilbert’s Hotel. Imagine there’s an infinite hotel, in which every room is currently occupied. (There’s a very popular convention in town, apparently.)
You show up, asking for a room. “Sorry, they’re all occupied,” says the innkeeper—who’s probably a little frazzled, what with the infinite calls for room service coming in.
“Yes,” you say, “but just ask everyone to move down one room, and there will be space.”
It’s true. The person in Room 1 goes to Room 2. The person in Room 2 goes to Room 3. The person in Room 136,004 goes to Room 136,005. And so on. Everybody has someplace to go.
Now, whether it’s good manners to show up at a hotel and demand that a literally infinite number of people suffer a minor inconvenience just so you can avoid trying the Best Western down the road—that’s another matter. You’re probably a bad person for making this request. But you’re a fine mathematician.
Can we do this without inconveniencing an infinite number of people? Unfortunately, no. For example, if you take Room 1, and send that guy packing, he’s going to have the same problem as you: every room is now occupied. As long as you’re shuffling a finite number of rooms, normal math applies, and somebody will be left out. Your infinite problem demands an infinite solution.
It gets crazier. What if you’re coming straight from the Infinite Man March, and you’ve got an infinite line of new friends in tow? Can the hotel hold you all now?
This time, each current guest needs to look at their room number, double it, and move to that doubled room. So the chap in Room 1 moves to Room 2. The gal in Room 2 moves to Room 4. The fellow in Room 3003 moves to room 6006. And so on.
At this point, which rooms are filled? Only the evens. So you and your infinity friends fill the odd rooms, and voila! Problem solved.
(Unless you’re the bellhop in charge of setting out the continental breakfast; then, the problems have just begun.)
At this point, infinity is perhaps beginning to feel like a squishy, flexible, silly-putty sort of idea. Infinite lists? They’re the same size. Infinity plus one? Still the same size. Infinity plus another infinity? Yup, still the same size. It raises the provocative question: Are all infinities the same size? Tune in next week for the answer.
Oh, who am I kidding? The answer is no. Some infinities are much bigger than others.
Tune in next week for why.