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A “curious puzzle” from Raymond Smullyan:
Imagine a plane table of infinite extent. Attached perpendicularly to the table is a rod of finite length, and above that, attached by a hinge, is a second vertical rod, this one infinitely long.
Operate the hinge. What happens? The infinite rod descends freely through the first 90 degrees, until it’s parallel to the tabletop. But it can’t go beyond this, because then at some point the solid rod would intersect the solid table.
Thus it’s impossible to “rest” an infinite rod on an infinite plane. “And so, you have the curious phenomenon of the hinged rod being supported at only one end!”