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U-Sets as a probabilistic set theory

A topos of presheaves can be seen as an extension of classical set theory, where sets vary over informational states, therefore it is a powerful and expressive mathematical framework. I introduce a suitable topos of presheaves where probabilities and probabilistic reasoning can be represented. In this way we obtain a mathematical definition of probabilistic sets. A valid and complete proof system, w.r.t. the intended semantics of probabilities, is described using the internal language of the topos.