www.carom-maths.co.uk Activity 1-17: Infinity

Many of the errors that mathematicians Of all the ideas you will meet in mathematics, the most elusive of them all is likely to be infinity. In no area are the preconceptions that we bring to thinking about the subject more likely to be adrift. Many of the errors that mathematicians have made down the years have come from not paying enough respect to infinity and its implications.

Consider the following sequence: What happens as the number of terms increases? The sequence clearly heads towards 0. What happens if we add the terms of the sequence? It seems clear to us that this heads towards 2. Indeed, the sum to infinity we can say IS 2.

So we have an infinite number of numbers that add to a finite number. Mmm… Does this explain Zeno’s Paradox? Achilles gives the tortoise a 100m head start in their race. By the time Achilles has run 100m, the tortoise has moved on say 1m. By the time Achilles has run this 1m, The tortoise has moved on... This argument can be repeated infinitely often. So Achilles can never overtake the tortoise. But we watch the race, and he does just that!

The error is to say that the sum of an infinite number of things must be infinite. If the things become infinitely small, then they can add to something finite. But... Just because they become infinitely small, they don’t have to add to something finite! for example...

Georg Cantor, Is infinity just ‘infinity’? Or might there be different types, different sizes of infinity? The mathematician Cantor spent much of his life thinking about infinity, and maybe he paid the price – his mental health was fragile. But he came up with two marvellous arguments that still shine a great light onto the idea of infinity today. Georg Cantor, German, (1845-1918)

What is the simplest idea of infinity that you can have? Maybe… 1, 2, 3, 4, 5… Is this the only infinity we can have? How about the infinity given by all the integers? …-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5… The looks to be a bigger infinity – but it turns out that you can rearrange all the integers to pair off perfectly with the positive integers.

Task: what is the rule for getting from n to m here? Can we find a similar rule for getting from m to n? We say that there is a bijection between the set of ns and the set of ms; and if there is a bijection between any two sets, then they are of equal size. What about the rational numbers? Surely this is a bigger set than the counting numbers!

So between every two rationals there is another rational… that’s not true for the counting numbers. But actually, the set of rational numbers is no bigger than the counting numbers. Suppose we arrange the rational numbers like this:

That means we can count them like this:

So once again, it seems that the infinity represented by the counting numbers is the one possessed by the rational numbers. Maybe this infinity is the only one then? What about… the infinity given by all the numbers between 0 and 1? Cantor discovered a wonderful argument here. Suppose that the numbers between 0 and 1 can be written out in a list as decimals; that is, suppose the infinity we are dealing with is the same as that of the counting numbers.

Cantor then said, suppose I take the following number: and then I change each digit for another, any other.

This new number will be between 0 and 1, but where will it be in the list? It cannot be at number n, say, because it differs with the nth number in the list at the nth digit. The only conclusion we can come to is that the number is not in the list, and so the list is incomplete… which means that the infinity of the real numbers is bigger than the infinity of the counting numbers.

‘Saying ‘the CH is true’ is consistent of standard mathematics.’ For a long time a big question in mathematics was ‘Is the Continuum Hypothesis true?’ The CH says there is an infinity between that of the counting numbers and that of the numbers between 0 and 1. Kurt Godel, Austrian-American (1906 –1978) ‘Saying ‘the CH is true’ is consistent with the axioms of standard mathematics.’

‘Saying ‘the CH is false’ is consistent of standard mathematics.’ Paul Cohen, American (1934 - 2007) ‘Saying ‘the CH is false’ is consistent with the axioms of standard mathematics.’

So Godel and Cohen showed mathematics works perfectly well (in a non-contradictory way) whether we assume this infinity between the counting numbers and the infinity of numbers between 0 and 1 exists, or whether we assume it doesn’t. So we now have two different versions of mathematics to work with – with the Continuum Hypothesis being true, and with it being false. I did say infinity was tricky!

With thanks to: Wikipedia, for another excellent article. Carom is written by Jonny Griffiths, hello@jonny-griffiths.net