Some Infinities Are Bigger Than Others

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We all use the term infinity, but what does it really mean? What does it represent, and why is it important?

Well, infinity represents something that is endless, or boundless, for example the digits of pi (π). Since pi never ends, it’s infinite. Pretty simple, right?

Let’s talk about sets and infinities. Sets are just a group of objects, like a few numbers or even some people. For right now, let’s consider the set of whole numbers (0 and onwards). Since this set of numbers continues forever, it is infinite in nature.

So, whole numbers are infinite. What about integers, which include both negative and positive numbers? Since this doesn’t end either, it’s infinite as well. But aren’t there more integers then whole numbers? Surely, since whole numbers are a subset (part) of integers, there have to be more integers.

But wait.

Since both have an infinite number of numbers, then an easy way of checking if they are of the same size is to check if each whole number can be mapped to an integer, checking if the two are bijective.

Before we get into that, let’s first try a smaller example. If I have a Set X of numbers 1, 2, 3 and 4, and a Set Y of A, B, C, D, the mapping would look like this:

Picture taken from Wikipedia.

Since every number of Set X is mapped to every number of Set Y, we can say that they are of the same size.

Now, if we repeat this procedure for integers and whole numbers, what does our mapping look like? Can we map each whole number to an integer?

It turns out that we can, because both have an infinite number of numbers, meaning that for every number from the Whole Number Set there is a number that can be mapped to the Integer Set.

Therefore, the number of integers and and whole numbers is the same!

Seems weird, doesn’t it?

Now, what about fractions and whole numbers. Surely, there are more fractions than whole numbers. Try the mapping for yourself, and you will find that no, even the number of fractions and whole numbers is equal!

If you want to try listing all the fractions, you can use this wonderful method created by Georg Cantor called Cantor’s diagonal argument.

So, are there no infinities that are bigger than others? Are all infinities equal?

To answer that question, let’s look at irrational numbers.

Irrational numbers are those numbers which cannot be accurately expressed as a fraction, like pi (yes, 22/7 is accepted as a fractional value of pi, but it is an approximation, and isn’t accurate).

Now, even these are infinite in nature, so can we try mapping them to integers or even fractions?

This would make you think that irrationals are equal in number to the fractions as well.

But…

As shown above, you can list all the fractions. However, you cannot list all the irrationals!

Here’s why. Imagine that you have a recurring decimal, or any irrational number for that matter, and write it down (to about 10 digits). For example,

3.1415926535…

Now, let’s take a random number, say 5. If this number is present in the first place of your decimal, write 1, and if not, write 2. For example,

3.1415926535…

would become:

3.2221222121…

So, if we continued this process forever (infinitely!), then we would come up with a whole new irrational number!

This proves that we cannot list all irrationals, making them uncountable while other sets like fractions are countable. Since countable is always less than uncountable (for obvious reasons!), the infinity of irrationals is larger than the infinity of fractions (or rationals).

There we have it! Some infinities are indeed bigger than others! Mind-boggling, isn’t it?

Student. Teenager. Human.