Why can't a Hilbert curve be used to put the real numbers into a listable format?

Question

There's a very good chance this question will make absolutely no sense, as my understanding of Hilbert curves is very superficial. But let me explain where my question is coming from.

From my understanding, a Hilbert curve can completely fill a plane, meaning it runs through every single point on that plane. This seems like to me like a very closely related concept to finding a way to list all of the real numbers, as both creating a list of the reals and completely filling a space with a curve seem impossible for the similar reasons. Yet one turned out to be possible, so in my head at least that would imply that the other should be as well.

Again, my understanding of Hilbert curves is too superficial to construct a very specific question, so the best I can hope to do is outline a general description of the feeling I'm trying to get at, and hope that more educated users are able to pick up on what I'm thinking and explain what I'm failing to consider.

Edit: I think what the real question I'm trying to get at is, why are these two concepts not both impossible for the same reason?

Answer 1

The Hilbert curve gives a pairing between $\mathbb{R}^2$ and $\mathbb{R}$. This means you can list all pairs of real numbers if you can already list all real numbers. It doesn't help you with the latter if all you can do is list the natural numbers.

Answer 2

$\newcommand{\Reals}{\mathbf{R}}\newcommand{\Nat}{\mathbf{N}}$Probably the two questions look "the same" because each amounts to "Does there exist a mapping from one set onto a 'much larger' set?" Specifically,

• Does there exist a surjection $f:\Nat \to \Reals$, i.e., a function from the natural numbers to the real numbers that takes every real number as a value? (Answer: No, e.g., by the diagonal argument.)

• Does there exist a surjection $f:\Reals \to \Reals^{2}$? (Answer: Yes, e.g., a Hilbert curve.)

That the answers differ leads us to conclude the cardinality of $\Nat$ is smaller than the cardinality of $\Reals$, but the cardinality of $\Reals$ is equal to the cardinality of $\Reals^{2}$. Perhaps that's the truly surprising fact.