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cumulant-nimbus comments on 0 And 1 Are Not Probabilities - Less Wrong

34 Post author: Eliezer_Yudkowsky 10 January 2008 06:58AM

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Comment author: cumulant-nimbus 11 January 2008 02:24:12AM 0 points [-]

What do you mean by "infinite set atheism"? You are essentially stating that you don't believe in mathematical limits -- because that is one of the major consequences of infinite sets (or sequences).

If you don't believe in those... well, you lose calculus, you lose the density of real numbers, you lose the need or understanding of man events with probability 0 or 1, and you lose the point of Zeno's Paradox. -- Janos is spot on about measure zero not implying impossibility. What is the probability of a golf ball landing at any exact point? Zero. But it has to land somewhere, so no one point is impossible.

Impossibility would mean absence from your sigma algebra. What's that you ask? Without making this painful, you need three things for probability: an idea of what constitutes "the space of everything", an idea of what constitutes possible events out of that space which we can confirm or deny, and an assignment of numbers to those events. (This is often LaTeX'ed as (\Omega, \mathcal{F}, P).) The conversation here seems to be confusing the filtration/sigma-algebra F with the numbers assigned to those events by P.

Can we choose which we're talking about: events or numbers?