Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
Visualization 2001 Conference (Acm
Visualization 2001 Conference (Acm
Algorithmic Randomness of Closed Sets *
Journal of Logic and Computation
CiE'06 Proceedings of the Second conference on Computability in Europe: logical Approaches to Computational Barriers
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The notion of immune sets is extended to closed sets and $\Pi^0_1$ classes in particular. We construct a $\Pi^0_1$ class with no computable member which is not immune. We show that for any computably inseparable sets A and B , the class S (A ,B ) of separating sets for A and B is immune. We show that every perfect thin $\Pi^0_1$ class is immune. We define the stronger notion of prompt immunity and construct an example of a $\Pi^0_1$ class of positive measure which is promptly immune. We show that the immune degrees in the Medvedev lattice of closed sets forms a filter. We show that for any $\Pi^0_1$ class P with no computable element, there is a $\Pi^0_1$ class Q which is not immune and has no computable element, and which is Medvedev reducible to P . We show that any random closed set is immune.