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MagnetoHydroDynamics comments on Logical Pinpointing - Less Wrong
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How do you determine whether a physical process "behaves like integers"? The second-order axiom of induction sounds complicated, I cannot easily check that it's satisfied by apples. If you use some sort of Bayesian reasoning to figure out which axioms work on apples, can you describe it in more detail?
Well, it comes from the fact that apples in a bowl is Exclusively just that, as verified by your Bayesian reasoning. There are no other "chains" of successors (shadow apples? I can't even imagine a good metaphor).
So, now you in fact have that bowl of apples narrowed down to {0, S0, SS0, SSS0, ...} which is isomorphic to the natural numbers, so all other natural number properties will be reflected there.