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Quinn comments on Open Thread, September 30 - October 6, 2013 - Less Wrong
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I suppose the question is: What should you do if you're offered a bet on whether the dart will hit the target or not?
There's no way to avoid the question other than arguing somehow that you'll never encounter an immeasurable set.
For the same reasons you outline above, I'm okay with fighting this hypothetical target.
If I must dignify the hypothesis with a strategy: my "buy" and "sell" prices for such a bet correspond to the inner and outer measures of the target, respectively.