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by Sam Eisenstat 9 days ago | Jessica Taylor likes this | link | parent

Two minor comments. First, the bitstrings that you use do not all correspond to worlds, since, for example, \(\rm{Con}(\rm{PA}+\rm{Con}(\rm{PA}))\) implies \(\rm{Con}(\rm{PA})\), as \(\rm{PA}\) is a subtheory of \(\rm{PA} + \rm{Con}(\rm{PA})\). This can be fixed by instead using a tree of sentences that all diagonalize against themselves. Tsvi and I used a construction in this spirit in A limit-computable, self-reflective distribution, for example.

Second, I believe that weakening #2 in this post also cannot be satisfied by any constant distribution. To sketch my reasoning, a trader can try to buy a sequence of sentences \(\phi_1, \phi_1 \wedge \phi_2, \dots\), spending \(\$2^{-n}\) on the \(n\)th sentence \(\phi_1 \wedge \dots \wedge \phi_n\). It should choose the sequence of sentences so that \(\phi_1 \wedge \dots \wedge \phi_n\) has probability at most \(2^{-n}\), and then it will make an infinite amount of money if the sentences are simultaneously true.

The way to do this is to choose each \(\phi_n\) from a list of all sentences. If at any point you notice that \(\phi_1 \wedge \dots \wedge \phi_n\) has too high a probability, then pick a new sentence for \(\phi_n\). We can sell all the conjunctions \(\phi_1 \wedge \dots \wedge \phi_k\) for \(k \ge n\) and get back the original amount payed by hypothesis. Then, if we can keep using sharper continuous tests of the probabilities of the sentences \(\phi_1 \wedge \dots \wedge \phi_n\) over time, we will settle down to a sequence with the desired property.

In order to turn this sketch into a proof, we need to be more careful about how these things are to be made continuous, but it seems routine.



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