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magfrump comments on Approaching Infinity - Less Wrong Discussion
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If you stereographically project the real numbers onto the unit circle and use the metric inherited from R^2, then in fact (2) is very close to infinity.
If you use the arctangent to project the positive reals onto a finite set, then again I would guess that (2) is very close to infinity.
There's an old joke...
Some psychologists do an experiment. They put a mathematician and a plumber on one side of a room, and a beautiful woman on the other side. They say, "You can cross half the remaining distance in the room as many times as you like." The mathematician sighs, "I'd never reach her!" The plumber shrugs and says, "You'd get close enough."