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The Typical Constructible Object

Mathieu Hoyrup 1
1 CARTE - Theoretical adverse computations, and safety
Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods
Abstract : Baire Category is an important concept in mathematical analysis. It provides a way of identifying the properties of typical objects and proving the existence of objects with specified properties avoiding explicit constructions. For instance it has been extensively used to better understand and separate classes of real functions such as analytic and smooth functions. Baire Category proves very useful in computability theory and computable analysis, again to understand the properties of typical objects and to prove existence results. However it cannot be used directly when studying classes of computable or computably enumerable objects: those objects are atypical. Here we show how Baire Category can be adapted to such small classes, and how one can define typical computably enumerable sets or lower semicomputable real numbers for instance.
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Mathieu Hoyrup. The Typical Constructible Object. Computability In Europe, Jun 2016, Paris, France. pp.115 - 123, ⟨10.1007/978-3-319-40189-8_12⟩. ⟨hal-01396167⟩

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