
Re: Ariadne's thread
Posted:
Jan 2, 2012 3:26 AM


Let me return to the paradox encountered on the advanced level of RMP 41 and formulate it more simply.
Imagine a cylinder. Halve the height and double the diameter. Halve the height and double the diameter. Halve the height and double the diameter. Go on like this forever. The cylinder turns into an ever thinner and larger disc whose volume, doubled with every step, increases beyond any limit. In the end you get a circle of an infinite diameter and an infinite volume!
Back to the original cylinder. Double the height and halve the diameter. Double the height and halve the diameter. Double the height and halve the diameter. Go on like this forever. The cylinder turns into an ever thinner and longer rod whose volume, halved with each step, gets smaller and smaller. In the end you have a straight line of infinite length and zero volume.
A line has no volume, but how can a circle have one? The above circle has an infinite diameter and an infinite volume.
The lesson to be learned may be that the infinite is full of surprises. Our mathematical insights, powerful as they may be, are but glimpses of a divine knowledge.

