1AM is not a good time to discover there’s more than one infinity

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The other night I was living my usual rockstar life: eating cereal whilst watching videos on YouTube. I started off with logic puzzles – with mixed results – and ended up watching an explainer of the ‘infinite hotel paradox’. I’ve embedded the video at the bottom of this post, but the basic premise is this:

The infinite hotel has an infinite number of rooms. It also has infinite guests, which means that every room is filled. So if another customer arrives at the front desk asking for a room, they’ll have to be turned away, right?

If you happen to be reading this post in the middle of the night, go to bed now. This quickly becomes one hell of a rabbit hole.

At one point in the video, the narrator casually mentions that there is more than one type of infinity. Sorry, what?

How can there be any more than one type of infinity? It’s infinity. It is infinite – surely it has no limitations? Which means you only need one of it, unlike – say – a car. The reason there are so many different types of car is because not every car suits every purpose. Imagine the advert for the first car that tried to meet everyone’s needs:

Getting the kids to school will be no trouble with the new seven seater Ford infinity. Thanks to its front, rear, four-wheel-drive automatic manual gearbox with snow treaded tyres with added smoothness, you will be able to make the school run no matter whether your children attend an institution located in the middle of the suburbs, at the top of the mountain, or on an iceberg surrounded by penguins. The engine is located in the front, middle, and rear, leaving storage space under the bonnet, in the passenger area, and at the back. This makes it incredibly convenient to store your professional corporate family work bicycles, golf shopping, and food clubs.

And so on…

So doesn’t infinity do everything? Apparently not. It gets even harder to wrap your head around when you realise that if you asked a mathematician ‘Doesn’t infinity do everything?’ they would get confused as to which infinity you were referring. As if you’d asked a colleague at work ‘What does Brian actually do here?’ and watched as they try to figure out which of the seven Brians on your floor you mean.

So then I watched another TED talk about infinity, and things got even more complicated. You see, the problem is there are different ways of counting to infinity (I still don’t know what the hell I’m talking about, so if anyone clever is reading this, I have probably made a lot of mistakes already – perhaps an infinite number of mistakes. Oh yeah, I went there.) If you counted to infinity in the obvious way (one, two, three, et cetera) you get one type of infinity. But if you counted to infinity using, for example, the Fibonacci sequence (one, two, three, five, eight, etc) then by the time you reached infinity, you would have used fewer numbers. So what you have is not only two types of infinity, but one that is technically ‘smaller’ than the other.

Or, to use my own example, you can think about infinity in terms of the physical world. Take the surface of a football, for example. It has no beginning or end, so it is infinite. Yet it is quite small, which would surely mean that compared to a beach ball (which also has an infinite surface area) it has a smaller level of infinity. I think this means that if you had a series of spheres, each one marginally bigger than the one before it, you could count infinity in terms of traditional numbers. The smallest sphere could be Infinity One, and the next Infinity Two, and so on. Eventually you could reach Infinity Infinity.

This all occurred to me at 1 o’clock in the morning. You won’t be surprised to learn that sleep did not come easily to me that night.

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