The universal semimeasure is "computably approximable from below", aka (lower) semicomputable, meaning that you can computably list out all the rational numbers below a given value assigned by the measure. The probabilities assigned by a logical inductor are computably approximable reals, which (confusingly) is weaker than lower semicomputable; it just means that you can compute a sequence of rationals that converges to the real, with a possibly uncomputable convergence rate.
I'm not an author but this property with respect to a universal 'semimeasure' holds in the limit as time goes to infinity, as do most stated properties of logical inductors. The fact that they're computable is academic.
This is right, but perhaps misleading; most of the properties are "asymptotic", meaning that they may take an extremely long time to hold, but they hold at finite times. For example, "provability induction" says that if you have a (polytime computable) sequence of sentences phi_n, all of which happen to be provable (possibly with fast-growing proof lengths), then P_n(phi_n) limits to 1. This means that on day n the logical inductor P is confident of phi_n, even though phi_n may take much longer than n steps to prove.
The key is that the generator of such sequences has limited resources; once the inductor has learned the (implicit or explicit) program generating these sentences (perhaps by simulation, as in Solomonoff induction), it can apply that reasoning to the individual sentences; e.g. if it believes it's verified the correctness of the generator, that is enough justification to believe the sentences being produced.
This is in the vein of "prediction using ensembles of experts" methods such as SI, with a twist that the experts are traders, not forecasters; they don't have to have opinions on everything the logical inductor has to predict, the traders just have to point out particular ways that the logical inductor is being silly (and then the logical inductor corrects those problems).
Yeah. It's still surprising to me, though; P_n can predict extremely long-running computations, even ones with a much longer runtime than P_n, at least as well as any quickly computable "pattern". (The algorithm in the paper uses a (roughly) double-exponential-time algorithm to predict arbitrarily long-running programs, in a way that can't be improved upon by any polytime computable method.)