We do it all the time. An index is just indicative that there is a mapping (a function), usually from the integers. However we don't use the subscript notation when indexing by a continuum due to the discomfort you describe.

The point is that we need some way to deal with objects that are inherently infinite-dimensional.

You can look at use cases for an index, and see how well they hold up.

Asking where the smallest greater number (next number) is no longer makes sense.

Taking two numbers and asking whether one is greater than the other still makes sense. (and hence also whether they are equal)

Taking two numbers and asking how far separated from each other still makes sense.

You may already observe some uses for indexes in programming that don't use all of these properties of an index. For example, the index of a hash set "only cares about equality", and "the next index" may be an unfilled address in a hash set.

Well there's no law against it.

Okay I suppose the axiom of choice is somewhat necessary to make it make sense. But only because otherwise such an indexed object may fail to exist.

Anyway arbitrary indexes are useful, you often end up doing stuff like covering a space by finding a covering set for each individual point. And then using compactness to show you only need finitely many to cover the whole space. It is doable without uncountable indices, but it makes it very difficult to write down.

I think getting hung up on words (in this case index) in mathematics is a trap. They are often stretched to their breaking point and you just kind of go with the flow.

> When I use a word,’ Humpty Dumpty said in rather a scornful tone, ‘it means just what I choose it to mean — neither more nor less.’

> ’The question is,’ said Alice, ‘whether you can make words mean so many different things.’

> ’The question is,’ said Humpty Dumpty, ‘which is to be master — that’s all.

In mathematics it is the author's privilege.

I think that’s why the author put “vector” in quotes. I kind of imagine it as an ephemeral, infinite list where for some real, when we use that real value as an index into our “vector”/function, we get the output value as the item in this infinite, ephemeral list.

I think the only thing that matters is that the indices have an ordering (which the reals obviously do) and they aren’t irrational (i.e. they have a finite precision).

Imagine you have a real number, say, e.g. 2.4. What stops us from using that as an index into an infinite, infinitely resizable list? 2.4^2 = 5.76. Depending on how fine-grained your application requires you could say 2.41 (=5.8081) is the next index OR 2.5 (=6.25) is the next index we look at or care about.

I could be misunderstanding it, though.

A vector is always a vector -- an element of something that satisfies the axioms of a vector space. The author starts with the example of R^n, which is a very particular vector space that is finite-dimensional and comes with a "canonical" basis (0,...,1,...,0). In general, a basis will always exist for any vector space (using the axiom of choice), but there is no need to fix it, unless you do some calculations. The analogy with R^n is the only reason the "indices" are mentioned, and I think this only creates more confusion.

> and they aren’t irrational (i.e. they have a finite precision)

No, if you want only rational "indices", then your vector space has a countable basis. Interesting vector spaces in analysis are uncountably infinite dimensional. (And for this reason the usual notion of a basis is not very useful in this context.)

> and they aren’t irrational (i.e. they have a finite precision).

I'm not sure if I'm misunderstanding what you mean by 'finite precision' but the ordinary meaning of those words would seem to limit it to rational numbers?

In practice you're always computing with finite precision. (Even computing with symbolic expressions is just a preliminary step to what's ultimately a numerical result with finite precision.) The whole point of real numbers is to abstract away from detailed considerations of precision, and figure out what happens if you only ever care about putting satisfactory bounds on the output and are willing to bound your input to the extent required.

What is the benefit of using R if it's really Q?

The idea of R is that it allows you to reason about things like "I need more than X input precision to achieve Y bound on my output". Just sticking with naïve computation in Q does not suffice for that.

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.

c.f.: fractional indexing

In ZFC set theory, indexed family over a set (possibly uncountable or even bigger), is just syntactic sugar for a function.

So let's say you have a set U (possibly uncountable). To say let {u_i}, i in I (another set, possibly uncountable) is equivalent to asserting existence of function f:I -> U, such that f(i) = u_i. Note that this does not even require axiom of choice, since you are allowed to postulate that a function exists.

Of course if I is uncountable you can't list the elements of I, but that is not important in this case.

This Has it's use. The continouus Fourier Transform is is based on that. You are asking what frequencies is this continouus signal made of. Time is normally defined as a real number in that context, but If you have a continouus time you need continouus frequencies to map time space to frequency space. You can think about an Index as a lego Block, that you need to construct Something.

The only difference of note, I think, is that you can't enumerate the elements. Instead of being able to say "for each element, ..." you'd have to say "for all elements, ...", like the example of vector length defined as an integral over the full number range.

To a mathematician “each” and “all” are synonyms.

The author is stretching an analogy, it's a price to pay for starting with R^3 as a motivational example. There is nothing in the general definition of a vector space that requires it's elements to be "indexed"

What do you understand “index” to mean here? To me, a family indexed by some set is mostly just a different notation for, and attitude towards, a function with domain the indexing set.

Consider a function on R as an |R|-dimensional vector...

Which defeats the purpose of thinking about functions as a vector space. It's all smokes and mirrors