Descriptive properties on admissible sets
type: Item
instance of: scholarly article
title: Descriptive properties on admissible sets
zbMATH Open document ID: 1228.03019
author: Vadim G. Puzarenko
published in: Algebra and Logic
publication date: 2011-01-24T00:00:00Z
review text: The author explores the relation between various descriptive properties on admissible sets, with focus on reduction (\(\mathsf{Red}\)), existence of universal \(\Sigma\)-functions (\(\mathsf{UF}\)), existence of \(\{0,1\}\)-valued universal \(\Sigma\)-functions (\(\mathsf{UF2}\)) and separation (\(\mathsf{Sep}\)). It is shown that admissible sets behave rather badly in that almost every relation between these properties may be realized on a certain admissible set. And in most cases, such a set can be chosen as a hereditarily finite superstructure \(\mathbb{HF}(\mathfrak{M})\) over a computable model \(\mathfrak{M}\). The paper is divided into four parts. In Part I, the relations between the following properties are stated: enumerability, uniformization, reduction, (\(\mathsf{UF}\)), (\(\mathsf{UF2}\)), separation, and extension. Part II focuses on (\(\mathsf{Red}\)): various examples of classical structures \(\mathcal{M}\) are given so that \(\mathbb{HF}(\mathfrak{M})\) does not satisfy the reduction property. Part III considers (\(\mathsf{UF}\)) and provides computable models \(\mathbb{HF}(\mathfrak{M})\) that disprove the implications \((\mathsf{UF})\to(\mathsf{Red})\) and \((\mathsf{Red})\to(\mathsf{UF})\), respectively. Finally, Part IV deals with (\(\mathsf{UF2}\)): computable structures of the form \(\mathbb{HF}(\mathfrak{M})\) are specified where (\(\mathsf{Sep}\)) and (\(\mathsf{UF2}\)) fail, and where (\(\mathsf{Sep}\)) and (\(\mathsf{UF}\)) fail, but (\(\mathsf{UF2}\)) still holds.
reviewed by: D. Probst
Mathematics Subject Classification ID: 03D60
zbMATH DE Number: 5840715
zbMATH Keywords: admissible set
zbMATH Keywords: descriptive property
zbMATH Keywords: universal function
MaRDI profile type: Publication
full work available at URL: s10469-010-9087-4
OpenAlex ID: W2026085252
cites work: Q4075450
cites work: Q5573961
cites work: Q4040892
cites work: Q4451652
cites work: Q5391560
cites work: Q4530728
cites work: Q5476783
cites work: Q5476771
cites work: Q4517267
cites work: S-Subsets of Natural Numbers
cites work: Products of locally cyclic, torsion-free groups
cites work: Q4936677
DOI: 10.1007/S10469-010-9087-4
description: scientific article; zbMATH DE number 5840715