It's finitely many pieces, but not in the way you think. Each supposed piece isn't something you could carve with a knife, even if we defined "carving" as a mathematical operation. It's just a set of points with coordinates that obey a certain rule.
In mathematics you can say things like "the set of all points in the sphere with irrational coordinates" and think of that as one "piece" of the sphere but it doesn't correspond to anything in the real world.
I'm not a mathematician either, but I think the consequence is just that mathematical points are rather weird and admitting any infinite set of points (however disconnected) as a volume produces weird results. What we think of as a "piece" isn't just an infinite set of points.
In mathematics you can say things like "the set of all points in the sphere with irrational coordinates" and think of that as one "piece" of the sphere but it doesn't correspond to anything in the real world.
I'm not a mathematician either, but I think the consequence is just that mathematical points are rather weird and admitting any infinite set of points (however disconnected) as a volume produces weird results. What we think of as a "piece" isn't just an infinite set of points.