Abstract
We study some properties of descriptive set theory which translate from the ideals of enumerability degrees to admissible sets. We show that the reduction principle fails in the admissible sets corresponding to nonprincipal ideals and possessing the minimality property and that the properties of existence of a universal function, separation, and total extension translate from the ideals to some special classes of admissible sets. We first give some examples of the admissible sets satisfying the total extension principle. In addition, we define a broad subclass of admissible sets admitting no decidable computable numberings of the family of all computably enumerable subsets. We mostly discuss the minimal classes of admissible sets corresponding to the nonprincipal ideals of enumerability degrees.
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Original Russian Text Copyright © 2005 Puzarenko V. G.
The author was supported by the Russian Foundation for Basic Research (Grants 01-01-04003; 02-01-00540) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh-2069.2003.1; MK-2452.2003.01)
Translated from Sibirski \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath} \) Matematicheski \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\imath} \) Zhurnal, Vol. 46, No. 1, pp. 185–208, January–February, 2005.
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Puzarenko, V.G. Computability in special models. Sib Math J 46, 148–165 (2005). https://doi.org/10.1007/s11202-005-0016-z
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DOI: https://doi.org/10.1007/s11202-005-0016-z