Infinite computations with random oracles

We consider the following problem for various infinite time machines. If a real is computable relative to large set of oracles such as a set of full measure or just of positive measure, a comeager set, or a nonmeager Borel set, is it already computable? We show that the answer is independent from ZFC for ordinal time machines (OTMs) with and without ordinal parameters and give a positive answer for most other machines. For instance, we consider, infinite time Turing machines (ITTMs), unresetting and resetting infinite time register machines (wITRMs, ITRMs), and \alpha-Turing machines for countable admissible ordinals \alpha

H-structures and generalized measurable structures

We study H-structures associated to SU-rank one measurable structures. We prove that the SU-rank of the expansion is continuous and that it is uniformly definable in terms of the parameters of the formulas. We also introduce notions of dimension and measure for definable sets in the expansion and prove they are uniformly definable in terms of the parameters of the formulas