A friend of Rachel's asked this question on Facebook: "Can anyone tell me why infinity is a useful mathematical concept?"

I started to write out an answer that Rachel could paste as a Facebook comment, but it was getting longer and longer (though not infinite): a couple of nice examples of infinity in math came to mind, but to really explain them would take more space than would make sense for Facebook. So I decided to write it out here. Disclaimer, if any mathematicians are reading: this was written very off-the-cuff and it's written for a general audience rather than a mathematical audience, so there might be some minor issues with notation and formality.

I. Unbounded Growth

'Infinity' can be an easy way of talking about a function's unbounded behavior. For example, you can ask for any function f(x) what the limit of f(x) is as x approaches infinity. What you're really asking is basically this: as 'x' gets higher and higher (and not just higher asymptotically to some fixed number, like it slowly gets closer and closer to 5, but higher in an unbounded way, meaning that it eventually passes 5, 10, 15, or any other given number), does the function's value get closer and closer to some number?

The function 1/x has a limit of 0 as 'x' approaches infinity: if you divide 1 by bigger and bigger numbers, the answer is closer and closer to 0. The function 1/x + 4 has a limit of 4 as 'x' approaches infinity. The function x^2 has no limit as 'x' approaches infinity. We can say that the limit of x^2 as 'x' approaches infinity is "infinite", or "equal to infinity", which would be shorthand for saying that it gets higher and higher in an unbounded way (again, this means that it will eventually get higher than any given number). Since there is no actual number 'infinity', it does not actually have a limit, of course. The same thing happens to the function 1/x as 'x' approaches 0 from the positive side (i.e. you start at, say, x=1, and let 'x' slowly get closer and closer to 0; the function then approaches infinity).

On the other hand, the function sin(x) also has no limit as 'x' approaches infinity, but for a different reason: it just oscillates forever between -1 and 1, so it doesn't get closer and closer to any particular number, and moreover, it doesn't grow in an unbounded way.

II. Set Theory

That's one usage of the concept of infinity in mathematics: to talk about unbounded growth. Another usage is in set theory. A 'set' is just a bunch of unique objects, in no particular order. So, {1, 2, 5} is a set, and that's the same set as {5, 1, 2}. Both sets have size 3, because they have three things in them (in this case, the things are numbers, but they can be anything). But you can also have infinite sets: the set of all natural numbers, or counting numbers, i.e. {1, 2, 3, 4, ...}, is clearly infinite, since no matter how high you count, you can count one higher. So it can be useful to talk about whether a set is finite or infinite.

One way things get interesting is that there is actually a notion that some infinite sets are bigger than others: this is the distinction between what's called a 'countably infinite' set and an 'uncountably infinite' set. A set is countable if you can list all of its elements: you name one element first, then you keep listing them, and the rule is that you have to eventually hit any given number of the set.

So, the set of natural numbers is countable: if you give me any natural number, (e.g. one billion), I can tell you for sure that you'll eventually reach it in the list {1, 2, 3, 4, ...}. Also, the set of all integers (i.e. all positive and negative whole numbers, along with zero) is countable, since you can make the list as follows: {0, 1, -1, 2, -2, 3, -3, ...}. Say you have a positive integer, like 100, that you want to reach: it takes longer to get to it in the second list than in the first one, but you still hit it after a while, and you also hit all of the negative integers. An interesting and totally non-obvious result is that the rational numbers are also countable (showing this is not super-difficult but takes a little bit of work).

But not every set is countable. Georg Cantor showed in his famous Diagonal Proof that the set of all real numbers (this includes rational as well as irrational numbers) is NOT countable. The proof is very elegant and I will reproduce it here (although it's not strictly necessary for the purposes of this post, so feel free to skip it):

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1) Suppose that the set of all real numbers is countable. Then, since the set S of all real numbers between 0 and 1 is a smaller part of the set of all real numbers, S must also be countable.

2) Since S is countable, we have a list of all numbers in S. We can express each number in the list as an infinite decimal, so it would look something like this:

S = {

0.163042...,

0.689143...,

0.333333...,

0.250000...,

0.999999...,

...}

(Some notes: not all of the numbers will have repeating decimal representations, since not all of them are rational. So, the first two numbers, from what I've written of them, may not be rational. The third number, if the 3s continue repeating, is 1/3. The fourth number is 1/4, and the fifth number is equal to 1.)

3) Let us produce a number 'x' using the following rule. We will go down the diagonal digits from our list of S. For the first digit of 'x' (after the decimal point), if the first digit of the first number in the list is not a '3', then we will make the first digit of 'x' a '3'; if it is, we will make the first digit of 'x' a '4'. For the second digit of 'x', we do the same thing, but we look at the second digit of the second number in the list. And so on. So, given our list above, 'x' will look like this: 0.33433...: the third number has a '3' as its third digit, so we have the third digit of 'x' as a '4', but the other digits shown are '3's since the other numbers do not have '3's in those digits.

4) This number 'x' must be different from any given number in our list: from how we constructed it, we can see that it differs from the 1st number in its 1st digit, the 2nd number in its 2nd digit, and so on. Therefore, 'x' is not in the list. But the list is supposed to contain every number between 0 and 1, and 'x' is clearly a number between 0 and 1, so we have a contradiction!

5) Therefore, it's impossible to have a list of all of the numbers between 0 and 1, and by extension, it's impossible to have a list of all of the real numbers.

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This remarkable proof tells us that the real numbers are uncountable: we can't make a list of them. What these results about countability tell us is that the countable infinite sets, such as the natural numbers, the integers, and the rational numbers, all have the same size in some sense: even though there are more integers than naturals in ONE sense (since every natural number is an integer, but not vice versa), in ANOTHER sense there is the same amount of numbers in both of them, since we can just make a list of each set and line up the lists. But since it's impossible to make a list of the real numbers, in a sense, that infinite set is much bigger than these other infinite sets. So here we get an interesting idea of some infinities being bigger than others.

Those are the two biggest examples of talking about infinity that came to my mind, but I'm sure there are plenty of others.

I just tried to remember the uncountable many proof the other day and failed to! Now I can't escape it!

ReplyDeleteCantor's Diagonal Proof is one of my favorite proofs in mathematics.

ReplyDeleteSame but I think my favorite proof is probably closer to your least favorite proof in absolute terms.

DeleteThere is no infinity large enough to capture how many levels of infinity there are!

ReplyDeletei don't believe you're the real chuck knoblauch. can you please throw this baseball to first?

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