# To Infinity and Beyond

The Droste Effect: A visualisation of infinity. Source: http://www.flickr.com/photos/johs/420412473/

It was quite a few years ago when I first heard the line:* To infinity, and beyond*.

Of course, at that time, I just brushed it off as a catchphrase- what with me being a ripe old age of one. But now that I really think about it is seems a bit absurd.

Infinity is…well… infinite? How could you possibly go beyond infinity?

If I were to write down the following equation:

1 x **a** = **a**

I am certain you would be able to tell me that there are an infinite number of values which I can substitute in for **a**. Let’s call all the possible values for **a** “Set 1”.

Now, if I give you another set of numbers, “Set 2”, all of which satisfy the condition: “each number in the set must be even“. You could also say that this is an infinite set of numbers.

But now ponder this: **Which infinity is bigger?**

Well, both odd and even numbers (and numbers which can be considered neither odd nor even) can be inserted into the above equation 1 x a = a, whereas only even numbers fit into set 2. So the second set is smaller than the first, yes?

*Cue:* “But wait, I thought there was nothing any bigger than infinity? How is infinity bigger than infinity? How is infinity smaller than infinity?”

Now I am going to introduce a concept which might help make a little more sense of what I have just said: countable and uncountable sets. A countable set is one for which I can say there is a definite first, second, third, fourth… element. An uncountable set is one that doesn’t have this property.

If I look at my first set: if I call zero the first element of the set, what’s the second element? I could say 0.0000000000000000000000001? Or 1 x 10^{-200}?

Oh wait.

There really isn’t a definite second value?

I could move the tiniest imaginable bit higher up the number line than zero and there would still be a number that lies between this new position and my original starting point. From the definition I have given you above, this means that what I have defined in Set 1 is an **“Uncountable infinity”**.

If instead we look at set 2. Starting at zero, the second element in the positive direction is obvious to anyone who has any experience with odd and even numbers: it’s 2. Then the third element is 4, the fourth is 6, etc. This is evidently a countable set. Thus, this infinity is a “**Countable infinity”**.

Of course, the fact remains that infinity is unbounded. It continues going on forever and ever so it is not at all possible to go beyond it. So “To infinity and beyond” could quite easily have been replaced with simply: “To infinity”.

But I suppose that doesn’t really have as nice of a ring to it?

Thank you 🙂

This is such a readable post; it feels really short and snappy even though it covers a really cool but somewhat convoluted concept. Nice work 🙂

Check above again 🙂 The examples I used were of two infinities where one (the uncountable) is larger than the other

I’m not quite sure what you mean Levi?

Interesting, though I kind of wish you had done an example where one infinity was bigger than the other

Thank you, I definitely. That was the first thing most people told me when I told them what I would be writing on for my next blog entry: “If it doesn’t contain a Buzz Lightyear reference, count me out”.

Logic is a very cool area of mathematics!

Nice little bit of logic there, well explained. I like the idea of distinguishing between countable and uncountable infinity- it’s not something I have heard of before. This is a nice example of how logic can be used to make sense of a novel issue.

And I am going to admit, Buzz Lightyear was the first thing that popped into my head when I read that catchphrase!