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Presentability of groups of \(\Sigma\)-definable permutations over admissible sets. (Russian, English) Zbl 1016.03042

Algebra Logika 41, No. 4, 459-480 (2002); translation in Algebra Logic 41, No. 4, 254-266 (2002).
The author examines possibilities for generalizing the well-known Nurtazin result that the group of all computable permutations of natural numbers is not isomorphic to a constructive group. It is proven that the group of all \(\Sigma\)-permutations of an arbitrary locally countable, recursively listed, admissible set is not \(\Sigma\)-presentable over the set. An example is constructed showing that this group can be presentable provided that the local countability condition is dropped.

MSC:

03D60 Computability and recursion theory on ordinals, admissible sets, etc.
03C57 Computable structure theory, computable model theory
20A15 Applications of logic to group theory
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