Logic and probabilistic systems

Following some ideas of Roberto Magari, we propose trial and error probabilistic functions, i.e. probability measures on the sentences of arithmetic that evolve in time by trial and error. The set T of the sentences that get limit probability 1 is a Pi(3)-theory, in fact T can be a Pi(3)-complete set. We prove incompleteness results for this setting, by showing for instance that for every k > 0 there are true Pi(3)-sentences that get limit probability less than 1/2(k). No set T as above can contain the set of all true Pi(2)- sentences, although we exhibit some T containing all the true Sigma(2)-sentences. We also consider an approach based on the notions of inner probability and outer probability, and we compare this approach with the one based on trial and error probabilistic functions. Although the two approaches are shown to be different, we single out an important case in which they are equivalent.