The Schwarz paradox demonstrates that surface area is not a straightforward generalization of arc-length. Arc-length is defined for rectifiable curves &emdash; for any such curve, we approximate by line segments. The arc-length is the limit of the sums of the lengths of the line segments.
For surfaces, this definition breaks down. Rectifiable surfaces are well-defined, but the limit fails to be well-defined. Schwarz found two different sequences of approximations to the cylinder that converge to distinct values for the surface area.
Here are some papers that explain the paradox:
- an (old) paper by Benoit Mandelbrot, Length and area “anomolies”.
- a brief historical introduction, The Schwarz Paradox by Rickey.
- An Application of Abstract Nonsense to Surface Area by Lord.
While searching on the topic, I also found a nice historical work, A Panorama of the Hungarian Real and Functional Analysis in the Twentieth Century. It touches on the Schwarz paradox, and many other topics besides. (For example, it explains what the Sz. in Sz. Nagy stands for.)