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

Gill sahab wrote an article recently where he quoted from a lecture given by Richard Feynman. I’m a big Feynman fan and "Surely You’re Joking, Mr. Feynman!" is one of my all time favorite books. He was a superlative teacher who had this uncanny knack of making you see common things in perfectly extraordinary ways. Below is an excerpt from one of his books, "What Do You Care What Other People Think?", a letter written by Henry Bethe (Hans Bethe’s son) to Gweneth Feynman on her husband’s demise.

---

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 ou. Name a number."

"One million." A big number to start.

"Two million."

"Twenty-seven."

"Fifty-f our."

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 ’inifinity’."

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