I was originally going to talk about groups today but I want to save that for tomorrow. I feel like I can give it a much better once over once I’ve had some time to mull it over. But for today I want to talk about something interesting. When you think about infinity what do you think? Presumably you think about the unending sequence of numbers that never ends. There is no such thing as infinity + 1 because infinity itself is not a single value.
But you’ll remember that from your childhood. That concept of “my infinity is bigger than your infinity.” This is purely madness of course, there is no such thing as one infinity that is bigger than another.
Or is there?
I’ve linked to a video before about an infinite sequence of infinite sequences. But that’s not even what I’m talking about today. I’m talking about infinities that have a beginning and an end (another thing I’ve talked about before*). But even this isn’t what I’m talking about.
Imagine if you will an infinite sequence between 0 and 1.
This sequence carries infinitely many numbers. No matter how many numbers you count I’ll always be able to find a number that you haven’t found yet. Once you’ve counted every number you can possibly imagine. Having used the equivalent computing power of every atom in the universe. I’ll be able to tack on a single digit to the end of any number in your series and there is a brand new number. Repeat ad infinitum.
And yet, you could just as easily create an infinite sequence that is now, tomorrow, and forever larger than this sequence. An infinite sequence between the numbers 1 and 2.
Your infinity is bigger than my infinity now! Even though we both have infinitely many values that add up to infinitely large numbers. Every moment of every second from now unto infinity you will have a bigger infinity. But I could come along and pick the sequence of 2 to 3.
It’s peculiar to me that we look at counting from 1 to 10 as some kind of simple feat. And yet every human being that does it has counted across more values than have ever existed in the universe. Each day you leap over infinity infinitely many times. Yet we do it so effortlessly that we don’t even notice it. It’s a good thing that time itself isn’t infinitely divisible. But that’s another problem entirely […which I’ve also written about…].
*I can’t actually find this from searching the site! Perhaps I’ll write it since I like it. Though I’ve touched on it in this post already.