Infinity

Say you have an infinite number of camels walking through the desert and a little baby camel is born, how many camels are there now? Surely there are still an infinite number, and surely since we added 1 more camel this infinity is greater than the original infinity. But is it even possible for infinity to be greater than infinity? So confusing!

The biggest error most people make when talking about infinity is to think of infinity as a number. In reality, infinity is not a number, it is a concept. By definition infinity is a quantity without end. So if you take a number and you keep adding 1 to it, then you are going toward infinity, but you never actually hit infinity since infinity is not a number.

So since infinity is not a number, what happens when you ‘add 1’ to infinity. As we saw earlier if we keep adding one we still have an infinite amount. So by adding 1 we are still at infinity. Now looking at the number of camels, even though we added 1 camel to the group, we still have an infinite number of camels and so the ‘number of camels’ has not changed. It’s a weird concept to grasp, but this concept (created by Georg Cantor) helps us understand infinity.

So then we are naturally inclined to ask, can we ever make a certain infinity greater than another infinity? Weird concept to think about, but thanks to Cantor and a bunch of other super smart mathematicians, we now know the answer is yes!

This concept is created in Set Theory where we look at the number of elements in a set. Cardinality is just the number of elements in a set, and a set is just a collection of objects. So if you have a group of 10 people, then the cardinality of the group is 10. The cardinality of the number of months is 12, the cardinality of the number of weeks is 52, the cardinality of the set of birthdays is 366. Well how about the cardinality of a set like the natural numbers? The natural numbers are the numbers 1, 2, 3, 4… So what is the cardinality since it seems like there is no end to the number of items in the set.

This is where the concept of infinity comes into play. Since the number of items in the set of natural numbers never ends, there are an infinite number of them. We call this type of infinity, countably infinite. The reason we call it countable is because we as humans can sit there and count them. We may never be able to hit the end (since there is none), but we can count them. (We can first say 1, then say 2, then say 3, etc. And if you say 1 number per second for 70 years you’d hit 2,209,032,000 !)

Well are there more countably infinite sets? Yes, integers and rational numbers are also countable! We can count integers by counting them in this way: 0, 1, -1, 2, -2, 3, -3, etc. How about rational numbers?! There seems to be a crazy amount of rational numbers, and in between 0 and 1 there are an infinite number, so they must not be countably infinte, surely. But they are!

How did mathematicians find this out? They took the rational numbers (which are all integers and fractions) and put them into a grid like the following:

 $$\frac{1}{1}$$ $$\frac{2}{1}$$ $$\frac{3}{1}$$ … $$\frac{1}{2}$$ $$\frac{2}{2}$$ $$\frac{3}{2}$$ … $$\frac{1}{3}$$ $$\frac{2}{3}$$ $$\frac{3}{3}$$ … … … … …

Now in order to make it countable just go in diagonals. So our sequence would be: $$\frac{1}{1}, \frac{2}{1}, \frac{1}{2}, \frac{3}{1}, \frac{2}{2}, \frac{1}{3}, etc.$$. Then in order to get all the negative numbers, just switch in between the two (positive, then the same number negative like we did for the integers.)

Now how about the set of all numbers (aka real numbers, which is the rational numbers with the irrational numbers such as $$\pi$$) It turns out that the real numbers are actually not countable. We can see this because there is no way to sit and count the real numbers. For example let us start with the number 1. What number do we choose next? 1.01 is wrong, because 1.001 is less and is also a real number. But we can’t choose that since 1.0001 is also a real number, so no matter what number we choose there is no logical next number to choose since we can keep finding a smaller number closer to 1.

Wait, now we have 2 different types of infinity! Countable and uncountable infinities. The cool thing is that if you look further into set theory you’ll find out that uncountable infinity has a higher cardinality than countable infinity! So there are the same number of whole numbers as rational numbers, but there are more real numbers than any other type of number. So cool!

If you want further information on Set Theory to be able to see different types of infinity I recommend A transition to advanced mathematics by Douglas Smith, Maurice Eggen, & Richard St. Andre (Books/Cole 2001) Have fun counting!