Property and structure in constructive analysis
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Real numbers such as Dedekind reals or (quotiented) Cauchy reals (as opposed to Bishop-style Cauchy reals) do not admit a procedure for observing information such as the first digit of its decimal expansion, because, for example, there are no non-constant functions into observable types such as the booleans or the integers. We overcome this by considering real numbers equipped with additional structure, which we call a locator. With this structure, it is possible, for instance, to construct a signed-digit representation or a Cauchy sequence. Such constructions are reminiscent of computable analysis. However, instead of working with a notion of computability, we simply work constructively to extract observational information, by changing one axiom of Dedekind cuts from property into structure.
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