Probabilistic and Team PFIN-type Learning: General Properties

We consider the probability hierarchy for Popperian FINite learning and study the general properties of this hierarchy. We prove that the probability hierarchy is decidable, i.e. there exists an algorithm that receives p_1 and p_2 and answers whether PFIN-type learning with the probability of success p_1 is equivalent to PFIN-type learning with the probability of success p_2. To prove our result, we analyze the topological structure of the probability hierarchy. We prove that it is well-ordered in descending ordering and order-equivalent to ordinal epsilon_0. This shows that the structure of the hierarchy is very complicated. Using similar methods, we also prove that, for PFIN-type learning, team learning and probabilistic learning are of the same power.