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Kingreaper comments on Infinite Certainty - Less Wrong
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If you can make a statement every two seconds, you could actually stand up and do this. If I could get sponsorship to offset existential risk, I'd take this challenge on to actually stand up for the best part of a day and make 10,000 true statements with nary a false one.
I would however go for less variety than you if I wanted to be confident of winning this challenge. "My teeth are smaller than Jupiter. The Queen is smaller than Jupiter. A Ford Mondeo is smaller than Jupiter..."
Those statements aren't even approximately independent though, if Jupiter turns out to be really small, they're all true. That's why mine were so weird, the independence clause.*
*(they still aren't actually independent, but I'm >99.99% sure you couldn't make a set of statements that were)
However, it's possible to make a set of statements that are mutually exclusive, which might actually be a superior task "I am not the 11,043rd sentient entity ever to exist. I am not the 21,043rd sentient entity ever to exist, etc."