Imagine you book into a hotel but are then told that all the rooms are full. Should you walk away and look for elsewhere to crash for the night? Well, that depends if the hotel has infinitely many rooms or not!
This thought experiment was devised by mathematician David Hilbert (1862–1943), and it teaches us that infinities are very odd things indeed! What is an infinity? To describe something as ‘infinite’ is to describe it as being unbounded – without limit. Take the natural numbers, for example: 1, 2, 3, 4… These are infinite, or rather, there are infinitely many of them, because there is no last member in the series they form. If we were to list them all, our task would never be done, no matter how much time we had, because there would be no point where we would run out of more to reel off. Now, suppose our hotel has infinitely many rooms, each with a different number on the door. Even in the case that every room is already occupied, the hotelier can accept more guests. For, all they need to do is ask guest in room 1 to move to room 2, and ask guest in room 2 to move to room 3, and so on and so forth. No matter how far along this hotel’s corridor’s we go, there will never be a guest who will have no adjacent room to move along to, because there is no end to the rooms in the hotel! This seems paradoxical: the rooms are all full, but more people can always be accommodated, even if infinitely many more people turned up to the already-fully-occupied hotel! But alas, our minds are not infinite, and there is a limit to what we mere mortals can grasp! | |||