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You’ve probably heard that some infinities are bigger than others. Personally, when I first heard that, I was skeptical. At the time, I thought infinity was the biggest number, that it was everything or forever. I thought it was the biggest thing, and so there couldn’t be something bigger than it. But as it turns out, to be infinite is simply to be greater than all the natural numbers (0, 1, 2…). What this means is that there is not just one infinity, but in fact infinite infinities!
Before we get to the infinite infinities, let’s talk about what most people refer to when they say infinity: the countably infinite. One countably infinite set is, rather intuitively, the set of counting numbers (1, 2, 3…). There are indeed an infinite number of these since you will always be able to count to a higher number. What makes them countable is the fact that, given a finite amount of time, you will be able to count to any one of the numbers. The said finite amount of time could be longer than the universe has existed, but you will, eventually, reach your number.
What you might think is a “smaller” infinity then, is the number of even numbers. After all, there should be half as many even numbers as counting numbers. But, these two infinities are actually the same. You can match every single even number to its own counting number by matching it with whatever it is divided by two. So, while it might not seem like it, there is an even number for every counting number and so they are the same infinity.
Now, let’s talk about some infinities of actually different sizes. First, the uncountably infinite. A common example of the uncountably infinite is the real numbers. We will even take a small part of the reals, the numbers from 0-1, and prove that there are more of these than there are counting numbers. To prove that this infinity is uncountable, we can prove that you cannot match the real numbers with the counting numbers.
What we will do first is assume that there are countably infinite real numbers between 0 and 1, and then write down every single real number on top of each other, digit by digit, as seen in the diagram below, in countably infinite rows. Next, for the number in row one, take its digit in column one, add one (if the digit is a 9, subtract one), and then write it down. Then, for the number in row two, take its digit in column two, add one, and write it down. Continue on until you’re “done”, and you should now have a number. This number is different from the first real because of its first digit, it’s different from the second real because of the second digit, and so on. This number is different from every single one of the countably infinite real numbers you wrote down, and so this must be a new real number! But what counting number can you associate it with? You used all of them already. As such, this is a real number which cannot be put with a counting number, and so this infinity is greater than countable infinity.
You can also think of this a little more intuitively. I told you earlier that for a countably infinite set of numbers, you can, in a finite amount of time, get to any number. For the real numbers from 0-1, we start at 0, and then… Right. There’s not exactly a “next” place to go. Presumably it would be something like 0.00000 and some infinite number of zeros and then a one, but how many zeros? There is no answer. You can’t even count to the second number, much less ANY number in the set, and so it is uncountably infinite.
Now I’d like to return to the countably infinite. Countably infinite is not a number, but a description. The number version of this is referred to as Aleph Null (ℵ0). Aleph numbers refer to the cardinality of infinite sets of numbers, which is basically how many elements are in the set. Aleph Null is the first of these Aleph numbers, and like I said, refers to a countably infinite cardinality. Aleph One (ℵ1), on the other hand, is an uncountably infinite number. It refers to the size of the set of all ordinal numbers.
An ordinal number is similar to the cardinal Aleph numbers, but it is ordered. This means that it counts the order of numbers in a set, instead of how many numbers are in it. We can define one ordinal number (ω) as the first number greater than all the counting numbers. This number has the same amount of elements as Aleph Null, but it is ordered. What this means is that for Aleph Null, ℵ0+1=ℵ0, since it can simply rearrange all the elements. However, ω+1 does not equal ω! This is because it is ordered, and so the unit being added to the end has nowhere to go, no place in the set. As such, ω+1>ω. What is funny though, is that 1+ω does equal ω. This makes ordinal addition non-commutative, leading to some funny arithmetic.
You can indeed do arithmetic with these ordinal numbers, ω+1…ω+2…ω+3, and so on until you get to 2ω…3ω…4ω. Then you can keep going until you get ωω. These are the infinite infinities I referred to at the beginning. With all that in mind, you might ask, what type of infinity describes how many infinities there are? The answer to this is… none? It’s like trying to create a box of all boxes. For every box you try to create to capture all of them, you create a new box. The number of infinities is so unimaginably large that it is not even infinite, it is beyond infinite. Just like how the number of finite numbers is not finite, the number of infinite numbers is not infinite. It’s an entirely different realm of “thing”, that you can’t even call a number.