by **jjb2011** » Sun Sep 17, 2017 1:15 am

It was great to read a story in the Times and a leading article on Friday 15th September concerning the nature of mathematical infinities. It turns out that some infinities are bigger than others, and some infinities, although they look like they are different in size, turn out to be the same size.

A good example of an infinite set is the set of all natural numbers, 1,2,3,4,5........and so on. Clearly this set is infinite. But what about the set of even numbers, 2,4,6,8.......and so on. Clearly this set of numbers is infinite as well, but this set, at first sight,has only half the numbers as the set of natural numbers. So, is the infinity of natural numbers twice the size of the infinity of even numbers ?

It turns out that these infinities are the same size. The explanation is due to Cantor. For these two sets of numbers, the numbers can be paired up. so for example .....

1 --> 2

2 --> 4

3 --> 6

4 --> 8 and so on.

So for every number in the set of natural numbers there is a corresponding number in the set of even numbers. Because of the this correspondence of elements the two sets which are both infinite, have the same size. These infinities are countable and one countable infinite set is the same as any other countable infinite set.

The situation is different when we look at real numbers. Real numbers are decimal numbers ..... 0.537892562 for example. The interesting thing about real numbers is that if you take any two real numbers, there are an infinite number of real numbers between them.It doesn't matter how close you choose the initial two real numbers. So the infinite set of real numbers is infinitly bigger than the infinite set of natural numbers. The real numbers cannot be put into a list and are not countable.

JJB

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