We introduce a certain reducibility on admissible sets which preserves definable predicates. Some lattice-theoretic properties are given of the ordered sets of the classes of admissible sets equivalent under this reducibility. Furthermore, we give a transformation that assigns to each admissible set some hereditarily finite set and preserves the following list of descriptive set-theoretic properties (with account taken of the levels of a definable hierarchy): enumerability, quasiprojectibility, uniformization, existence of a universal function, separation, and extension. We introduce the notion of jump of an admissible set which translates the descriptive set-theoretic properties considered above to the corresponding properties lowering levels by 1.In [1] the notion of Σ-reducibility on admissible sets was introduced that preserves the Σ-theory. Its main merit, as well as demerit, is the preservation of the structural properties of an admissible set such as the height of an admissible set and the structure of elements. In this paper we study some reducibility that preserves the Σ-theory rather than the particularities. It is this reducibility that will be called Σ-reducibility here. The notion of Σ-reducibility was introduced in [2]. It turned out that to study the most important properties of this reducibility we can consider only the hereditarily finite sets, i.e., the inclusion-least admissible sets. We show that for each admissible set there is an equivalent hereditarily finite set preserving a series of descriptive set-theoretic properties; in particular, the reduction principle and the existence of a universal Σ-function. As a corollary of this transformation, we give a series of lattice-theoretic properties. The main result and its corollaries were announced in [3]. However, the signature estimation is improved here. A plenary talk on the conference "Mal cev Meeting-2004" was also dedicated to these results.The lattice-theoretic and structural properties of this reducibility were studied earlier [4][5][6][7]. It turned out that the reducibility coincides on countable admissible sets with the reducibility which was introduced in [8]. It acts on the classes of arbitrary admissible sets in the same way as the reducibility suggested in [9].We introduce the notion of jump of an admissible set which translates the descriptive set-theoretic properties in the corresponding properties lowering levels by 1.
This paper calculates, in a precise way. the complexity of the index sets for three classes of computable structures: the class of structures of Scott rank , the class , of structures of Scott rank , and the class K of all structures of non-computable Scott rank. We show that I(K) is m-complete is m-complete relative to Kleene's and is m-complete relative to .
A reducibility on families of subsets of natural numbers is introduced which allows the family per se to be treated without its representation by natural numbers being fixed. This reducibility is used to study a series of problems both in classical computability and on admissible sets: for example, describing index sets of families belonging to Σ 0 3 , generalizing Friedberg's completeness theorem for a suitable reducibility on admissible sets, etc.
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. This article continues the research of [1, 2] into the relations between various properties over the ideals of e-degrees and admissible sets. The methods we develop enable us to describe the computable properties of the minimal classes of admissible sets. In § 2 we define a property of Σ subsets similar to indiscernibility, applying it in § 4 and § 5 to studying the computable principles, and in § 3 to studying the existence problems for the Friedberg numberings on the admissible sets of minimal classes. In § 1 we collect the preliminaries as well as the main properties of minimal classes of admissible sets. 1480037-4466/05/4601-0148
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