How big is infinity?"asked Jo.
"Why do you ask?" "My book says, 'No other world was carried through the starry infinity on the backs of four giant elephants'. I wondered how big infinity is."
Terry Pratchett fans will recognise Jo's quote from Discworld. Pratchett has created an imaginary universe and also plays with scientific and mathematical ideas. I suggested we should think about infinity for homework. Next day, I was given 30 definitions. Most included words like boundless, endless, innumerable and even "see infinite". But what did they tell us?
It was my turn to ask an awkward question. "So, infinity is the biggest number you can think of?" This was agreed; but, I asked, just how big is that?
We worked our way up the scale, with plenty of examples - how many days in a year?. How many hours?
"And the biggest number there is?" They began to combine number names with no real idea of place-value.
"You can make your biggest number even bigger by adding one," I suggested.
"And then we could go on adding ones until we reach infinity," someone said. "But how will we know when we get there?" "We never stop. Infinity isn't a place, it's a way of saying that a sequence is never-ending. That's what infinity means."
This was a revelation to the class and made me wonder how many concepts we assume children understand because we do.
Try this test of adult understanding of infinity: think about the set of all even numbers (2,4,6...). Is it half as big as the set of all whole numbers (1,2,3...)?
Did you use your common sense and say, "Yes."? But wait. If you pair up every whole number with its partner in the even number set, like thisI ...neither will run out before the other!
As the mathematician Cantor, working on comparisons of infinite sets, said of one of his proofs, "I see it. but I don't believe it."