Having finitely many pieces is different from finitely many operations you would need to cut the pieces in real life.
For example, there is a curve that cuts the plane into just two pieces, but the line itself has infinitely small resolution [1] and hence you can't cut it. In fact, there is no segment of this line, no matter how small, that you can cut in a finite amount of time!
For example, there is a curve that cuts the plane into just two pieces, but the line itself has infinitely small resolution [1] and hence you can't cut it. In fact, there is no segment of this line, no matter how small, that you can cut in a finite amount of time!
[1]: http://en.wikipedia.org/wiki/Weierstrass_function