By Dr. Aaliya Mir
Mathematicians have just discovered a new infinity. One that does not follow any rules of Mathematics. It is so strange that it could change how we understand logic, numbers, even the universe itself. It sounds impossible, right? Infinity is supposed to be the biggest thing that there is. How can there be more than infinite?
Well, it turns out that not all the infinites are of then same size, some are actually bigger than others. To make sense of them, one mathematician (George Cantor) came up with a way to organize infinity. Something called the infinity ladder. In this article, we are going to climb it. Each rung reveals a new kind of endlessness, stranger and more powerful than the last. But this new infinity, it seems to live on a rung that doesn’t belong. One that could bring the entire ladder crashing down. This chaos has even sparkled a bit of a rebellion. Mathematicians who say that we should get rid of infinity altogether, they call themselves “ultrafinitists”. Their argument that infinity isn’t just confusing, it is misleading us, holding science back by letting us use quantities that could never exist in reality. And if they are right, it might mean something astonishing. That even beyond what we see, the universe itself could be finite. This is a strange story that we finally have reached the end of infinity. So how can one infinity be bigger than the other? To find out, we need to step inside one of the most interesting thought experiments in all of mathematics.
Hilbert’s infinite hotel:
The experiment was thought of by a German mathematician, that is David Hilbert, and it goes like this:
Imagine a hotel with an infinite number of rooms. Now it is a good day for the owners, they are fully booked. There are an infinite number of guests filling the infinite number of rooms. Suddenly, a person shows up looking for a room. Surely, with a infinite rooms, you must have space for one more, right? Luckily, the manager has a moment of inspiration. All the rooms are full, but this is an infinite hotel, so usual rules don’t apply. The manager asks the person in Room no. 1 to move to Room no. two, the person in Room no. 2 to move to Room no. 3, and so on. Every guest moves up a room no.. It is an infinite number of room changes, but this is an infinite hotel. So it is just another normal day at the office. The effect of all of these room changes is that there is now nobody in room no. 1. Thus the guest settles in there. Apparently, the manager spots a bur through the window, unfortunately, which doesn’t appear to be a usual kind of a bus. It’s an infinite bus with infinite number of people in it, expecting an infinite no. of beds at the hotel to sleep in. Can it be done?
Well, the manager didn’t become the manager of an infinite hotel for nothing. He has another trick up his sleeve. Now, this time, unlike before, every guest moves double whatever their room no. is. That means the guest in room no. 1 moves to room no. 2, the guest in room no. 2 moves to room no. 4, the one in room no. 3 to room no. 6, the one in room no. 4 to room no. 8 and so on. The effect of this is that all the existing guests are now in even numbered rooms, 2, 4, 6, … , leaving all the odd numbered rooms free. There are infinitely many odd numbers. So as each person steps off the infinite bus, they get assigned an odd numbered room. Once again the infinite hotel satisfies all its guests maintaining its illustrious infinitestar rating.
Now, what have these room shenanigans told us about infinity? The hotel is nothing but a short hand for the counting numbers. 1, 2, 3, 4, 5,…., as every room has a corresponding room number. So this sized infinity calculated by Hilberty is called as a Countable infinity. What the thought of Hilbert hotel experiment shows is that . Also, the same experiment shows that . It all hints at a trick that the mathematicians use to show that an infinity is actually a countable one. To do that, one simply has to show that it can be paired up with the counting numbers. Or in another words, you have to be able to fit everyone into the infinite hotel. An argument like we saw for the infinite bus can be used to show that the odd numbers are a countable infinity, for example, or the even numbers. But, not all infinities are actually countable. There are those that are so large that the infinite hotel would never have a room for them. Those are the uncountable infinities. Now once you establish that one infinity can be bigger than another, suddenly a different picture starts to appear. There is a hierarchy of infinities. And this hierarchy appears to be far richer and complex than we ever thought it would be. And this strikes other very foundations of mathematics. Mathematicians have been trying to understand infinities since the dawn of mathematics. The ancient Greek philosopher, Zeno(6:20)
~ Dr. Mir Aaliya, hail from Sopore. She has pursued Ph.D. in Mathematics from Lovely Professional University, Punjab. Currently working as a Guest Faculty at Govt. Degree College, Pattan.
