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Epictetus comments on 0 And 1 Are Not Probabilities - Less Wrong
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Comments (128)
In theory, if you could list every possible observation you could make, that will have a 1 probability. It would take infinite time, because the following class of outcomes:
has an infinite cardinality. I could get into how Godel means you can't even in principle describe all possible outcomes in a finite amount of space, even by referencing classes like I did, but I'll leave that up to you.
There was a suggested fix to your problem in the post, why isn't that good enough for you?
Sounds like he agrees that S has probability 1.
Note: I agree that the way he "proves" the claim is not very good. He basically tries to switch your intuition by switching the wording of the question. Not too rigorous.
When I say that the possibilities can be listed in principle, what I mean is that there some set S that contains them and make no reference to any practical problems with describing or storing its elements. Like the points and lines of geometry, it's a Platonic idealization.
Because talk of magical symbols is a good sign that the passage was meant to ridicule the use of infinity. The very next paragraph seeks to expunge such "magical symbols" from probability theory.
If he has a rigorous way to ground probability theory without 0 and 1, I'm fine with it. He seemed to be saying that he wishes there was such a way, but until someone develops one, he's stuck with magical symbols. He acknowledges all your problems in the end of the post.