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Computable Closed Euclidean Subsets with and without Computable Points

Stéphane Le Roux·Martin Ziegler
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Abstract

The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskii, Tseitin, Kreisel, and Lacombe assert the existence of NON-empty co-r.e. closed sets devoid of computable points: sets which are `large' in the sense of positive Lebesgue measure. We observe that a certain size is in fact necessary: every non-empty co-r.e. closed real set without computable points has continuum cardinality. This leads us to investigate for various classes of computable real subsets whether they necessarily contain a (not necessarily effectively findable) computable point.

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Stéphane Le Roux, Martin Ziegler. 2007-06-22. Computable Closed Euclidean Subsets with and without Computable Points. https://arxiv.org/abs/cs/0610080

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