Douwe Osinga's Blog: Infinite numbers

Friday, December 5, 2003

Hilbert's hotel has an infinite number of rooms and they're all full. A guy shows up and the desk clerk says: we're all full, but I'll find a room for you. He moves the guest in room 1 to room 2, the guest from room 2 to 3 etc. The new guest can go to room 1. Infinite + one = Infinite. If an infinite number of guests arrives, there is still room enough: move the guest in room 1 to room2, the guest in room 2 to room 4, the guest in room 3 to room 6 etc. All the uneven rooms are now free, an infinite amount of rooms. See Mark Pilgrims blog for more details.

A couple of days ago I wrote about the ungoogable numbers, numbers that Google doesn't know about and what would be the smallest. Well, I'm still working on this. Meanwhile I'd like to present the class of unknowable numbers.

Somewhere in the argumentation about infinity without a miss, the distinction between countable and not-countable infinite shows up. In Hilbert's hotel, there are an infinite number of rooms, but they're countable. That is, if somebody picks a room and I start counting all the rooms from one up, eventually I'll get to his room. Weirdly enough, this is not true for the real numbers between 0 and 1. You cannot create a list of all those numbers, not even an infinite long list. See for a prove Cantors Diagonals Argument.

Now consider the set of all numbers that can expressed in mathematical language. These numbers can be ordered (for example alphabetically) so they are definitely countable and infinite. Let's say somebody claims that this list actually contains all the numbers from 0 to 1 and is immune against the Cantors Diagonals Argument.

Our guy gives Cantor his list and Cantor starts calculating and then comes up with a number not on the list. 'But how did you calculate?' our guy asks. Cantor shows him. 'Ah. But your calculation is a mathematical expression, so it is on the list by definition', our guy says. It is not according to Cantors definition of his list of course. Who are we to trust?

It is a weird paradox. In the end I think we'll end up with a collection of unknowable numbers, numbers that exists but we can't name, then they would be countable.