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tut comments on Open Thread, September 30 - October 6, 2013 - Less Wrong
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Immeasurable objects are physically impossible. The actual target will be measurable, even if the way you came up with it was to try to follow the "instructions" that describe an immeasurable set.
Hmm. What is the exact length of your, say, pen? Is it a rational number or a real number... I mean the EXACT lengh...?
Note if the answer to the last question is "it is a real number", then it is possible to construct the bet as proposed by the OP.
Before you quote "Planck's Length" in your reply, there is currently no directly proven physical significance of the Planck length (at least according to Wikipedia).