I think Diffractor’s post shows that logical induction does hit a certain barrier, which isn’t quite diagonalization, but seems to me about as troublesome:
As the trader goes through all sentences, its best-case value will be unbounded, as it buys up larger and larger piles of sentences with lower and lower prices. This behavior is forbidden by the logical induction criterion… This doesn’t seem like much, but it gets extremely weird when you consider that the limit of a logical inductor, P_inf, is a constant distribution, and by this result, isn’t a logical inductor! If you skip to the end and use the final, perfected probabilities of the limit, there’s a trader that could rack up unboundedly high value!
There’s unpublished work about a slightly weaker logical induction criterion which doesn’t have this property (there exist constant-distribution inductors in this weaker sense), but which is provably equivalent to the regular LIC whenever the inductor is computable.[1] To my eye, the weaker criterion is more natural. The basic idea is that this weird trader shouldn’t count as raking in the cash. The regular LIC (we can call it “strong LIC” or SLIC) counts traders as exploiting the market if there is a sequence of worlds in which their wealth grows unboundedly. This allows for the trick you quote: buying up larger and larger piles of sentences in diminishingly-probable worlds counts as exploiting the market.
The weak LIC (WLIC) says instead that traders have to actually make the money in order to count as exploiting the market.
Thus the limit of a logical inductor can count as a (weak) logical inductor, just not a computable one.
I think Diffractor’s post shows that logical induction does hit a certain barrier, which isn’t quite diagonalization, but seems to me about as troublesome:
There’s unpublished work about a slightly weaker logical induction criterion which doesn’t have this property (there exist constant-distribution inductors in this weaker sense), but which is provably equivalent to the regular LIC whenever the inductor is computable.[1] To my eye, the weaker criterion is more natural. The basic idea is that this weird trader shouldn’t count as raking in the cash. The regular LIC (we can call it “strong LIC” or SLIC) counts traders as exploiting the market if there is a sequence of worlds in which their wealth grows unboundedly. This allows for the trick you quote: buying up larger and larger piles of sentences in diminishingly-probable worlds counts as exploiting the market.
The weak LIC (WLIC) says instead that traders have to actually make the money in order to count as exploiting the market.
Thus the limit of a logical inductor can count as a (weak) logical inductor, just not a computable one.
Roughly speaking. This is not quite an adequate description of the theorem.