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TheAncientGeek comments on Philosophy professors fail on basic philosophy problems - Less Wrong Discussion
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How is a process of reasoning based on an infinite stack of algorithms concluded in a finite amount of time?
You can stop recursing whenever you have sufficiently high confidence, which means that your algorithm terminates in finite time with probability 1, while also querying each algorithm in the infinite stack with non-zero probability.
Bingo. And combining that with a good formalization of bounded rationality tells you how deep you can afford to go.
But of course, you're the expert, so you know that ^_^.