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Cyan comments on The Absolute Self-Selection Assumption - Less Wrong
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Comments (38)
Observing a Geiger counter near a piece of radioactive material was one of the highlights of my undergraduate physics labs. And the time distribution of clicks is random in the same sense that the OP was using.
As Sniffnoy notes, the bigger problem is about the observation of an actual real number. Any observable signal specifying the instant at which the particle triggered the counter has finite information content, unlike a true real number. This includes the signal sent by your ears to your brain.
I shouldn't have mentioned pseudo-random number generation in the grandparent -- it's a red herring.