A certain reducibility on admissible sets

Abstract

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.

Rice sequences of relations

“…The importance of the role of S-reducibility in this context was previously shown by Stukachev [22,23]. Another independent definition of the jump is due to Puzarenko [24]. However, he defines only the jump of an admissible structure, and does not deal with structures in general.…”

“…The notion of S-reducibility between abstract structures was introduced independently by Khisamiev [28] and Stukachev [22], and it is based on the notion of S-definability of a structure inside an admissible set by Ershov. Stukachev [22,23] and Puzarenko and co-workers [24,27] showed the importance of this reducibility and compared it with various other reducibilities between structures (including ≤ w ). The definitions of earlier studies [22,28] used, of course, the notions of S-definability rather than rice.…”

Fixed points for the jump operator

“…In view of the Σ-reflection principle, the class of all Σ-predicates on A coincides with Σ 1 (A). Relations between classes of definable predicates, as well as elementary properties of these classes, were discussed in [1]. Definition 2.…”

“…Recall that Wf (A) is an admissible set for each structure A of KPU in a relational language (see, e.g., [7,Prop. 2.7.4] We proceed to cite basic notions and assertions from [1]. Let A be a KPU-structure.…”

Descriptive properties on admissible sets

“…n corresponding to the admissible sets can be taken computable [6].) For transitions 7-9, such admissible sets are missing.…”