# Richard Feynman's incomplete proof

The following letter is from Richard Feynman's book What Do You Care What Other People Think?. It is written after his death by Henry Bethe, a son of 1967 Nobel Prize winner Hans Bethe, to Feynman's wife. In the letter, Feynman is called by his nick name Dick. Henry wrote about Dick's proof of "there are twice as many numbers as numbers", which I suppose is his way of saying "the union of two countable sets is also countable". It seems that he only shows the existence of the map

${\mathbb N} \owns n\mapsto 2n \in 2 {\mathbb N}$

But, I think this proof is incomplete unless he also shows that there is another map

${\mathbb N} \owns n\mapsto 2n-1 \in 2 {\mathbb N}-1$

and that both of the two maps are one to one and onto. Am I right?

Dear Mrs. Feynman,

We have not met, I believe, frequently enough for either of us to have taken root in the other's conscious memory. So please forgive any impertinence, but I could not let Richard's death pass unnoticed, or to take the opportunity to add my own sense of loss to yours. Dick was the best and favorite of several "uncles" who encircled my childhood. During his time at Cornell he was a frequent and always welcome visitor at our house, one who could be counted on to take time out from conversations with my parents and other adults to lavish attention on the children. He was at once a great player of games with us and a teacher even then who opened our eyes to the world around us.

My favorite memory of all is of sitting as an eight- or nine-year-old between Dick and my mother, waiting for the distinguished naturalist Konrad Lorenz to give a lecture. I was itchy and impatient, as all young are when asked to sit still, when Dick turned to me and said,

"Did you know that there are twice as many numbers as numbers?"

"No, there are not!" I was defensive as all young of my knowledge.

"Yes there are; I'll show you. Name a number."

"One million." A big number to start.

"Two million."

"Twenty-seven."

"Fifty-four."

I named about ten more numbers and each time Dick named the number twice as big. Light dawned.

"I see; so there are three times as many numbers as numbers."

"Prove it," said Uncle Dick.

He named a number. I named one three times as big. He tried another. I did it again. Again. He named a number too complicated for me to multiply in my head.

"Three times that," I said.

"So, is there a biggest number?" he asked.

"No," I replied. "Because for every number, there is one twice as big, one three times as big. There is even one a million times as big."

"Right, and that concept of increase without limit, of no biggest number, is called 'infinity'."

At that point Lorenz arrived, so we stopped to listen to him. I did not see Dick often after he left Cornell. But he left me with bright memories, infinity, and new ways of learning about the world. I loved him dearly.

Sincerely Yours,

Henry Bethe

• I suspect Feynman was saying what he was saying, and was making no pretence of giving a mathematically rigorous proof. – Lord Shark the Unknown Jul 29 '18 at 10:13
• @LordSharktheUnknown, in that case, why does being able to double every number mean "there are twice as many numbers as numbers"? – Aki Jul 29 '18 at 12:41
• Feynman was talking to an eight year old. What do you want? – Gerry Myerson Jul 29 '18 at 12:43
• I don’t know what the question is. – Randall Jul 29 '18 at 15:46
• @GerryMyerson, Do you think Feynman was successful in his attempt to get across the idea of "there are twice as many numbers as numbers" to the boy by just showing every number can be doubled. I feel something is missing in this supposedly nice episode in which "uncle" Dick gave some idea of infinity to a eight-year-old boy. – Aki Aug 2 '18 at 12:40

I think your characterization of the argument as "the union of two countable sets is also countable" is inaccurate. It seems to me that Feynman's point is that the map $$n\mapsto \left\lfloor\frac n2\right\rfloor$$ is two-to-one at every point.
Consider the analogous situation for finite sets: Let $C$ and $D$ be finite sets and let $f:C\to D$ have the property that for each $d$ in $D$, there are exactly two elements of $C$ that are mapped to $d$.
When either $C$ or $D$ is finite, the existence of such an $f$ is both necessary and sufficient for $C$ to have twice as many elements as $D$. For example, the faces of a cube can be painted red, yellow, and blue so that each color is used on two faces. (Details left as an exercise.) So a cube has twice as many faces as there are primary colors. No such coloring is possible for, say, the sides of a heptagon, or an octagon.
The same argument, taken with both $C$ and $D$ as the set of all numbers, suggests that there is an important sense in which there are twice as many numbers as there are numbers—as in fact there is.