Comment by mb7733
9 hours ago
> In a real-indexed vector, that notion doesn't apply. It's "infinity plus one" all the way down: whatever real value you pick to start with, x, there's no delta small enough to add to it such that there's no number between x and x+d.
Just to clarify, uncountability isn't necessary for this. It's true for the rational numbers too, which are countable.
Yes. Indexes in infinite sets are counterintuitive, and real numbers even more so.
The famous counterexample to all of this sort of thinking is Hilbert’s hotel, which I’m sure you know but want to point it out for people who haven’t seen it before because it’s pretty mind-blowing when you first encounter it.
Say you have a hotel with an infinite number of rooms numbered 1,2,3,… and so on and they are all occupied. A guest arrives- how do you accommodate them? Well you ask the person in room one to move to room 2, the person in room 2 to move to room 3, and in general the person in room n to move to room n+1. So every existing guest has a room and room 1 is now free for your new guest.
Ok but what if an infinite number of prospective guests arrive all at once and every room in your hotel is full. How do you accommodate them? Still no problem. You ask the guest in room 1 to move to room 2, the one in room 2 to move to room 4, and in general the guest in room n to move to room 2n. Now all your existing guests still have a room but you have freed up an infinite number of (odd-numbered) rooms for your infinite number of new guests to move into.
These are all countable infinities, and Cantor showed that if the number of rooms in your infinitely-roomed hotel is ℵ_0, then the number of real numbers is 2^ℵ_0, which is obviously quite a lot more.