Computable Closed Euclidean Subsets with and without Computable Points

Details Computable Closed Euclidean Subsets with and without Computable Points Computable Closed Euclidean Subsets with and without Computable Points

2006-10-13

arxiv.org/abs/cs/0610080v5

Computer Science

Logic in Computer Science

Included helpful remarks of J.Miller and of X.Zheng

Scientific paper

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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