> it is impossible to import things into something that has infinite volume because by definition there is no outside to import things from.
If you cut a 2d plane with a line, then both halves are infinite, yet both have "outside". You can repeat this and get infinite (aleph 0) number of divisions, each still as infinite as the initial plane.
Same way you can cut infinite 4d universe into countably many infinite 4d universes.
> not all of them are populated; therefore only a finite number are
By same logic: not all integer numbers are even, therefore only a finite number of numbers are even.
Wouldn't you get aleph_1 divisions, if, by methods left as exercise to the reader, making a cut for each real number? For example, how many angles are there in a circle? But the real kicker is, that taking the 2D surface of a ball, certainly the area is finite even if there are no bounds at all.
Addendum: The set of cuts would then include all points of the 2D-surface. So in principle, as the number of cuts approaches real infinity (pun intended; if you might call it that) the area left between the cuts approaches zero. That's certainly not infinite. But this cannot be done with only circle and straight edge - thus in no system of only two dimensions, pretty much by analogy - yikes - I mean the apparent isomorphism between Cartesian and polarized coordinate-systems is the only one I know in 2D.
Certainly, cutting is the arch example of proportional rationing, so the word alone implies rational numbers. Then, cutting is the act of removing a set of points from one set. So, the same construction over the real line but circled only a quarter around a midpoint is effectively removing two quandrants, half of the circle. So then you could say you have two infinite sets, but exactly because they don't touch (interact with) each other.
> it is impossible to import things into something that has infinite volume because by definition there is no outside to import things from.
If you cut a 2d plane with a line, then both halves are infinite, yet both have "outside". You can repeat this and get infinite (aleph 0) number of divisions, each still as infinite as the initial plane.
Same way you can cut infinite 4d universe into countably many infinite 4d universes.
> not all of them are populated; therefore only a finite number are
By same logic: not all integer numbers are even, therefore only a finite number of numbers are even.