> I don't immediately see how either of those would work for uncountable infinite sets, but it may be possible.
We can think of an N-dimensional space as a grid of (hyper)cube "cells":
- Cantor's pairing function traces diagonal lines through the top-right quadrant of a (countable) 2D grid
- We can extend Cantor pairing to the whole (countable) 2D plane by joining the diagonals of each quandrant (they meet at the axes). The resulting line spirals out from the origin, in a diamond shape.
- We can extend the 2D plane to higher dimensions, again by joining diagonals when they meet at the axes. This results in a diamond-shaped "shell" spiralling out from the origin.
The above gives us a 1D line which visits every cell one after another. Next we associate each unit-interval on our line with the cells we're traversing, i.e. the interval between 0 and 1 is associated with the 1st cell; the interval between 1 and 2 is associated with the 2nd cell; and so on. (We could also jump between positive and negative, like [0, 1), [-1, 0), [1, 2), [-2, -1), ... to include the negative half of our 1D line as well)
Finally, we use a space-filling curve, like a Z-curve, to map each interval to the volume of its associated grid cell. This works for uncountable intervals and cell volumes.
The result would be a line which starts at the origin, and gradually spirals outwards. From a distance, the line appears to form N-dimensional diamond-shaped 'shells'. If we look closer, we see each 'shell' has jagged edges, since it's formed of N-dimensional grid cells. If we look even closer, we see that the line traces a z-curve through each grid cell, filling its volume.
The above works for any pairing function and space-filling curve, although I've written about cantor pairing at http://chriswarbo.net/projects/procedural/cantor.html and z-curves at http://chriswarbo.net/projects/procedural/z.html :)