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.
