Comment by eucyclos
9 hours ago
I'm probably ignorant of how indexes work at a nuts-and-bolts level, but intuitively this seems like a good idea for certain situations. E.g if we want to keep entries in a specific order but don't know ahead of time how many entries will be added between two existing ones. House numbers in areas with a lot of development are an example of the kind of problem this seems ideal to solve, when there's a clear 'order' based on geography but no clear limit on the number of addresses that could be added 'between' existing addresses.
I think you're still describing a countably infinite set: there's a bijection between the natural numbers and the set of houses.
One way to think about it is that, even though you're defining an index that permits infinite amounts of subdivision, from any given house there's always a "next house up" in the vector: you can move up one space.
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.
> 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.
That’s kind of how I understand it as well.