Would it be possible to extract all the points in the area and still have positive area? I'm reading Apostol's Calculus. Given the axioms


  


I got a silly doubt. A point is a measurable set with zero area, according to the axioms. Suppose I have a square with $a(S)=16$ and a point with $a(P)=0$, then I can use $(3)$ from the axioms above and subtract one from another. 

Wouldn't that imply that I could apply $(3)$ repeatedly, remove all the points in $S$ and still have $a(S)=16$ even when $S$ has no points and should have $a(S)=0$?
No, because no matter how many times you apply (3) removing one point, you'll only have removed finitely many points.
I'm completely out of my league here, so bear with me. But I recently viewed a video from vsauce that seems to suggest that theoretically what you describe might actually be possible and would end up creating The Banach–Tarski Paradox