Mixing mathematical abstraction with the material world leads to apparent paradoxes as, for example, in case of curves of infinite length that enclose finite areas or surfaces in 3d of infinite area that enclose finite volumes. I came across another paradox in that series in a chapter by Raymond Smullyan in a book in honor of the 90th birthday of Martin Gardner.
Imagine that you have a solid plane table with a finite rod bolted perpendicular to the table. To the top of this finite rod is hinged one end of an infinite rod. The hinging allows the infinite rod to move up and down, but the curious thing is that the rod cannot possibly move down because both it and the table are solid, and therefore the rod cannot pierce the table. And so, you have the curious phenomenon of the hinged rod being supported at only one end.
By the same token, if the rod is extended to a line, the latter won't be able to move either up or down as moving one end up means moving the other end down. And so it appears that the line is in a perpetual equilibrium balanced on the tip of the vertical rod. This fact remains true regardless of how wildly the mass of the infinite line is distributed.
- R. Smullyan, Memories and Inconsistencies, in A Lifetime of Puzzles, A K Peters, 2008
- What Is Infinity?
- What Is Finite?
- Infinity As a Limit
- Cardinal Numbers
- Ordinal Numbers
- Surreal Numbers
- Infinitesimals. Non-standard Analysis
- Various Geometric Infinities
- Paradoxes of Infinity