Infinity and beyond

We mathematicians know that there is more than one type of infinity, and that some infinities are infinitely larger than other infinities. Non mathematicians struggle to understand how this can be possible. This is a simple thought experiment that demonstrates this. I thought of it many years ago and have used it with children and adults ever since. People seem to understand it.

I call this a proof, but when I say proof, it probably doesn’t have the rigour of Cantor’s diagonalisation proof.

​The proof

Imagine that you have a ten sided dice with the numbers 0 to 9 on the faces.
On a piece of paper write down 0. (That’s zero point).
Roll the dice and write down, after the decimal point, the digits you get. You are gradually writing the decimal for a number between 0 and 1. You are going to keep doing this forever.

What is the chance that you will roll a fraction?

Well, to get a fraction you need to either have a decimal that terminates, in which case you need to roll zeros for a very long time, or you need to have a recurring decimal, so that’s repeating the same pattern over and over for ever.

You can think about and discuss this prospect. Let’s say you have something that looks like a fraction after 10,000 rolls. This is promising, but to get the next digit correct you have a 1/10 chance. To get the next two you have 1/100 etc. The probability diminishes very quickly to become almost zero.

Technically the chance of getting a fraction tends to zero as the digits mount up. In other words, in all the numbers our dice can roll there are almost no fractions. The infinite number of real numbers is so much greater than the infinite number of rationals that the chance of picking a rational is almost zero.

 

Some questions arise about this “proof”.

Can we consider an experiment that is technically impossible? You cannot roll a dice an infinite number of times. I would suggest that the same argument can be applied to Cantor’s proof. He envisages writing down infinitely long decimals, doing this an infinite number of times and then creating an infinitely long diagonal. At least I have only one infinite process.

What do you mean by “almost” zero? Well technically this means an event has probability zero but is not impossible. It is just so unlikely that almost certainly cannot happen. In schools they like to equate probability zero with impossible, but it is not quite the same thing.

Can we have a ten sided dice? There is no regular platonic decahedron! The answer is a ten faced prism that you roll.

This all reminds us of the infinite number of monkeys who will type the complete works of Shakespeare.

“To be or not to be. That is the gezornenplaz” Bob Newhart

“The quality of mercy is not strnen” The Mekons first album

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