Numberings and definability in the hereditarily finite superstructure of a model
type: Item
instance of: scholarly article
title: Numberings and definability in the hereditarily finite superstructure of a model
zbMATH Open document ID: 0924.03055
author: A. N. Khisamiev
published in: Siberian Advances in Mathematics
publication date: 1999-10-28T00:00:00Z
review text: A model \(M\) is said to be \(\Sigma\)-definable over an admissible set \(A\) provided it is isomorphic to a model whose basic set is a quotient of a \(\Sigma\)-subset of \(A\) modulo a \(\Delta\)-equivalence and all predicates and functions are \(\Sigma\)-definable modulo this quotient along with their complements. If a model possesses a \(\Sigma\)-definable numbering, it is called \(\Sigma\)-enumerable or internally enumerable. The author proves a natural criterion for \(\Sigma\)-definability of a model in the hereditarily finite superstructure over another internally enumerable model. He also proves that, if a countable superatomic Boolean algebra is \(\Sigma\)-definable over a hereditarily finite superstructure of an internally enumerable model, then its ordinal type is also \(\Sigma\)-definable over it. The same result is proven for reduced Abelian groups and their Ulm types. The article under review is a translation of Vychisl. Sist. 156, 44-58 (1996; Zbl 0902.03030).
Mathematics Subject Classification ID: 03C57
Mathematics Subject Classification ID: 03D45
Mathematics Subject Classification ID: 03D60
zbMATH DE Number: 1353848
zbMATH Keywords: relative recursiveness
zbMATH Keywords: numeration
zbMATH Keywords: admissible set
zbMATH Keywords: constructivization
zbMATH Keywords: finitely generated algebra
zbMATH Keywords: \(\Sigma\)-enumerable model
zbMATH Keywords: \(\Sigma\)-enumeration
zbMATH Keywords: \(\Sigma\)-definable numbering
zbMATH Keywords: \(\Sigma\)-definability
zbMATH Keywords: hereditarily finite superstructure
zbMATH Keywords: internally enumerable model
zbMATH Keywords: superatomic Boolean algebra
zbMATH Keywords: ordinal type
zbMATH Keywords: reduced Abelian groups
zbMATH Keywords: Ulm types
reviewed by: Andrei S. Morozov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1353848