Now follow the blue line, omitting doubles, and assign subsequent positive integers to every cell you visit. You just numbered all the rational numbers! Since they are numbered, they all have a pair in the natural numbers set and therefore the amount of them is the same (another name for ℵ 0 -sized sets stems from that proof method: we call them countable since you can count their elements using natural numbers). Thus, the number of rational numbers can also be written as ℵ 0 . Now, onto the greater infinities. Imagine a number line and two rational numbers as close to each other as possible. Notice that there are still a lot of numbers between them that are not rational. As you get closer and closer you discover an infinite amount of real numbers between each pair of rational numbers, which there is, of course, an infinite amount of. It’s like infinity squared! This exclamation is, in fact, a rather close description of the number of real numbers. There are a lot of them, even more than rational numbers. The proof above relayed strongly on intuition, but there is, of course, a more rigorous one, which I will not write down here, but it involves imagining a list of all the rational numbers and creating new numbers out of them that are not on the list. The amount of real numbers is called continuum. This brings us to one of the most interesting problems of the set theory: the continuum hypothesis. The hypothesis simply asks whether there is an infinity that is bigger than ℵ 0 , but smaller than continuum. And yet it landed on David Hilbert’s list of the twenty-two important open questions in mathematics, and generally puzzled mathematicians for a very long time. The interesting part of the hypothesis is that it cannot be proven. It cannot be disproven either. It was mathematically proved to be unprovable, and then mathematically proved to be undisprovable, which is just weird. It just hangs somewhere in-between, neither true nor false, and mathematicians try to come up with new ways to crack it (since it is not impossible that with completely new tools the hypothesis could be one day settled). But we do not need to consider problems this complicated to have some fun with infinities. There are many problems that are easier and still fun. Imagine a hotel that has an infinite amount of numbered rooms, each housing a mathematician. Now, what shall the hotel’s management team do if a new mathematician arrives? They certainly should not just dismiss him or make him sleep on a random couch – mathematicians are sensitive beings; the poor guy could catch a cold or something. The hotel’s staff needs to find an empty room in a hotel in which every single room is already taken. The resolution is actually surprisingly simple – just make every mathematician move to the room next door (if he was in room one, he should go to room two, if he was in room two, he should go to room three, and so on). Now room one is free! A swift and mind-bending solution, involving merely the pain of infinite mathematicians having to change rooms all of a sudden, which they would surely be willing to endure for a colleague in need. … Now what happens if a coach with infinite mathematicians arrives and they all want a room? What if an infinite number of coaches arrive, each with an infinite number of mathematicians inside? The world of infinity problems seems to be truly endless and I do encourage you to look yourself for some of the wonderful and often weird things that happen in there. The Hilbert’s Hotel – the absolute classic of number theory 1 2 3 4 5 6 Is there something in between?
12
Made with FlippingBook - professional solution for displaying marketing and sales documents online