> points not on those lines form an uncountable set What do you mean? A countable set of lines cuts the screen into a nonsensical "set of regions", you can't d

> points not on those lines form an uncountable set What do you mean? A countable set of lines cuts the screen into a nonsensical "set of regions", you can't do it. The infinite pixel scanline is the set of all finite paths (it's all *terminating binary strings), not the set of all infinite paths. Since the set of cutlines is dense, you can actually name all regions, so the only sensible way to assign pixels is to assign a pixel to every line, not to the space between lines (since there is no nonzero space between lines)