To Infinity & Beyond

Romario Catania | BSc. Statistics & Computer Science 2nd Year.

I’m sure that you’ve seen that knocked-down 8 symbol before and you know what it represents.  But have you ever stopped to think about what it really means?

            Those who study maths or statistics have several study-units which are about infinite quantities or infinite processes.   When I first encountered these topics I had a hard time coming to grips with them, but they were the most beautiful ideas that I’ve been exposed to at university so far. 

            Let’s start with the simplest example of an infinite object.  Your teachers and your parents probably told you when you were little that you can never run out of numbers.  Let’s throw out all the fractions and all those weird numbers like e, π  or -5 and concentrate on the counting numbers.  You know, like 0, 1 and 89 - what you’ve been using since you were three years old.  (Mathematicians like to call the collection of these as the set of natural numbers.)

            You would agree that the set of even numbers (0, 2, 4, 6, 8….) are a subset of the counting numbers (0, 1, 2, 3, 4, 5, 6, 7, 8 …).  What if I told you that there are just as many counting numbers as even numbers?  This would probably not make sense to you because if you choose, say, all the numbers less than 10, there are 5 even numbers (0, 2, 4, 6, 8) and 10 counting numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).  This holds for any number you might choose, not just 10.

            Imagine if you put all the numbers one after the other in a list:

                        0  1  2  3  4  5  6….

            and then after that you put all the even numbers:

                        0  1  2  3  4  5  6  7  8  9  10….
                        0      2       4      6      8       10….

            You can see the set of even numbers as what happens when you double the distance of each number from 0.  When you do this transformation you aren’t adding any more numbers to the list, so you’d have to agree that the two (infinite) lists have the same number of items.

              Another way to see this is by giving an index number to each even number:

            1st 2nd 3rd 4th 5th 6th 7th …
            0    2     4    6     8    10   12…

           

  You can refer to 8 as the 5th even number, and use the number 5 to refer to the value 8.  Since there is a reversible operation to switch between these two sets, you can see these two sets as two different ways to represent the same thing.  (Mathematicians would say that these two sets are isomorphic.) So if they can represent the same thing, they would obviously have the same size. 

            Using this idea any infinite collection of things which can be ordered in some way have the same size as the counting numbers. If you’re creative, you can even see that there are just as many fractions as there are counting numbers!

            This is just skimming the surface when it comes to talking about infinities. The subject is full of paradoxes.  There are infinities that are greater than others, and infinities which you wouldn’t expect to be of the same size.  We aren’t used to thinking about infinite quantities, which is why the ideas are so odd to us.  However they are important because they underpin much of the mathematics that we use today such as probability and calculus.