I wanted to give you a tiny, little glimpse of the sensations that I experience daily on working in Set Theory. I want to take you on a stroll in the paradise that Cantor provided us. I am going to describe some really simple infinite set, and the only thing you have to do is to try to imagine it. Sometimes the mental image just don't come: don't worry, just keep thinking about it, and return back to it in another moment. Also, take your time. Don't just say "Yes, I understood" and go on, try to explore with your mind every nook and cranny of the infinite set.
And have the right attitude. Which attitude, you say? This one:
This advertisement is incredible. It manages to introduce effortlessy a nice amount of infinite sets. But can you really image infinity times infinity? Try it now. I'll wait.
What took you so much? So let's see if my description is the same as yours.
This is one.
This is two (yeah, I know, stay with me).
And so on. Just adding one on the right, you are making +1.
Once you finished the numbers, you have the first infinite set! Congratulations! We call it ω. (Infinity! like the girl said).
Now, you know how to do ω+1: after ω, add one point.
Well, there is not much space there, let's go to a new line.
And now add another one (ω+2).
And so on, until you finished the numbers again. We have ω+ω! (Infinity plus infinity, like the guy said).
Now add another point, and continue. We have ω+ω+ω, i.e. three times infinity.
Go on, with four times infinity, five times infinity, and so on. At the end we have ωxω (Infinity times infinity! like the other girl said).
Great, this was not that difficult. So let's go deeper. I want you to imagine ωω, i.e. Infinity to the power of infinity! Let's do it visually.
Visualize again ω.
Now, between each two points, add ω. We have ωxω, i.e. ω2.
Do it again: between any two points in ω2, add ω. We have ω3.
Of course I am not going to make a drawing for that, don't be silly. This is the moment when we have to use just our minds, the screen does not have enough resolution for that. Now do it again (add one ω for each point in ω3), and again (ω4), and again! In the end you have in front of you ωω!
Let's try another way. Imagine you are on a road, with pickets on the side of the road. The first picket you see is one meter high. You start counting them: one, two, three...
When you counted them all, one picket two meters high appears. Then again the short ones, start counting again: one, two, three...
counted them all the second picket two meters high appear. Continue like this, and a third, a fourth will appear, and so on. When you counted all the two meters high pickets, one three meters high appears!
And then the small ones again, etc. After the second batch of two meters high there is a second three meters picket, after the third batch a third, and so on. Once all the three meters high are finished, here it is one of four meters. Continue like this, and you will have passed ωω pickets.
Let's try even another way. Imagine a tally counter. But instead of having just numbers between 0 and 9, every disk has ALL the numbers. And instead of having finite disks, it has infinite disks. There: the tally counter can count exactly ωω numbers.
Homework! How to visualize Infinity to the power of infinity to the power of infinity? And Infinity to the power of infinity to the power of infinity to the power of infinity? And Infinity made to the power of infinity infinite times? Have fun!
One personal note: it is disarming how practically all the children manage to think about infinity plus one, like above pretty much. And yet many times they are stopped by adults, that tell them that you cannot do infinity plus one, because infinity is infinity. Of course you can! This reminds me of a schoolmate at kindergarten that once told me that 100 is the biggest number. I asked her "What about 103?" and she answered "It does not exist". In the same vein, my teacher at primary school told me that you cannot do 4-6. "Isn't it -2?" I asked. She told me "That does not exist, and that operation is FORBIDDEN". I am still recovering from that. So please, if you are reading this, think of the children! Don't limit their fantasy, don't close them the gate to Cantor's paradise!
I think that the people that make the most fascinating jobs have still inside them the kid that they were. Every astronaut is driven by the "childish" fascination of the stars, every zoologist by the passion kids have for animals, and what about paleontologists and dinosaurs! Well, si parva licet, I am still playing the game of "who says the biggest number".
And I am winning.
The story does not end here. You can find an update at this page. Also, did you know that there is a dance song about all of that?