In our latest episode of My Favorite Theorem, my cohost Kevin Knudson and I talked with Bates College mathematician Adriana Salerno about one of the prettiest theorems there is, Georg Cantor’s proof that there are more real numbers than there are whole numbers. You can listen here or at kpknudson.com, where there is also a transcript.
Salerno talked about a somewhat painful personal connection she has with this theorem. In college, one of her professors told her she shouldn’t be a math major anymore because she made a “mistake” writing up a proof of the theorem.
To understand what it means for there to be more real numbers than whole numbers, we need to know how to measure the size of infinite sets. We do that by defining two sets to have the same size if they can be paired up perfectly. Each element of the first set has exactly one buddy in the second set and vice versa.
If the set of real numbers were the same size as the set of whole numbers, we could list the real numbers in a list the way we can list the whole numbers. There would be a real number that came first on the list, a real number that came second, and so on. Cantor’s diagonal argument shows that no matter how you form your list of real numbers, it will be incomplete. Some real numbers—in fact, infinitely many of them—will not appear on the list.
To prove this result, you start with a proposed listing of real numbers. Then produce a number not on the list by changing the first digit of the first number, the second digit of the second number, and so on. When Salerno wrote her proof down on an exam, she wrote it like this:
Let the first number of the list have digits A1, A2, A3, and so on. Let the second number have digits B1, B2, B3, and so on. The third number is C1, C2, C3,.... You get the idea.
She proceeded to change the digits A1, B2, C3, and so on, capturing the main big concept of Cantor’s proof. The professor gave her no credit. His reasoning: The alphabet only has 26 numbers, so she only proved that a list of 26 numbers is incomplete.
Pardon my language, but that’s baloney. There might be better ways to write up this proof, but she had the basic idea. Frankly, it’s kind of hilarious to me that the professor would be fine with the idea that we could write down arbitrarily many digits of an infinite number of numbers if we were actually trying to carry out this procedure but didn’t have the imagination to believe we could figure out ways to keep track of digits once we ran out of letters in the alphabet. There was a whole Dr. Seuss book about this! If we’re not being practical anyway, why assume we can’t enlarge the alphabet? It’s kind of like watching an episode of Star Trek and having no problem with warp drive or the holodeck but finding Dr. Crusher’s near-instantaneous wound-healing technology implausible.
Salerno says she was too stubborn to listen to the professor’s recommendation to quit math, and clearly she ended up making it, but it’s a real bummer that she had to be that stubborn. I’m glad that for the most part my math professors encouraged me even when my proofs were a bit awkward or unwieldy and recognized that minor bookkeeping problems didn’t mean I had no understanding of the concepts.