Countably decomposable admissible sets
type: Item
instance of: scholarly article
title: Countably decomposable admissible sets
zbMATH Open document ID: 0558.03017
DOI: 10.1016/0168-0072(84)90006-X
author: Menachem Magidor
author: S. Shelah
author: Jonathan Stavi
published in: Annals of Pure and Applied Logic
publication date: 1984-01-01T00:00:00Z
review text: This paper is an important contribution to model and definability theory for infinitary languages unifying work on this subject done by various authors. An admissible structure \({\mathcal A}\) is defined to be countably decomposable if its domain A is the union of countably many of its elements and \({\mathcal A}\) satisfies a certain decomposability property DP. The main result of Part I states that for structures with the DP and theories \(T\subseteq {\mathcal L}_ A\) such that T is \({\mathcal A}\)-r.e. and a countable union of members of A, (a) the set of consequences of T in A is \({\mathcal A}\)-r.e., (b) if every \(T_ 0\subseteq T\), \(T_ 0\in A\) has a model then T has a model and (c) the least ordinal not pinned down by T is a member of A. This result includes all the known theorems on \(\Sigma_ 1\)-completeness, \(\Sigma_ 1\)-compactness, non-definability of well-orderings, etc. for various admissible structures. (The proof uses the method of open games.) In Part II, it is shown that under the assumption \(V=L\) the authors' result is the best possible in a certain sense. The notion of \(\Sigma_ 1\)-compactness is strengthened to stable \(\Sigma_ 1\)-compactness, and in Part III it is proved that \(L_{\alpha}\) is stably \(\Sigma_ 1\)-compact if and only if it satisfies the right decomposability conditions. The paper contains an appendix by \textit{L. Harrington}: ''Admissible ordinals are not immortal''.
reviewed by: K.Gloede
Mathematics Subject Classification ID: 03C70
Mathematics Subject Classification ID: 03D60
Mathematics Subject Classification ID: 03E45
zbMATH DE Number: 3889512
zbMATH Keywords: definability
zbMATH Keywords: infinitary languages
zbMATH Keywords: admissible structure
zbMATH Keywords: decomposability
zbMATH Keywords: compactness
zbMATH Keywords: Admissible ordinals
MaRDI profile type: Publication
cites work: Q4075450
cites work: Hanf numbers for fragments of L\(_{\infty \omega}\)
cites work: On the Semantics of the Constructible Levels
cites work: Q5565095
cites work: The Suslin-Kleene theorem for \(V_\kappa\) with cofinality(\(\kappa)=\omega\).
cites work: Aspects of constructibility
cites work: A partition calculus in set theory
cites work: Q3215216
cites work: Uncountable Admissibles I: Forcing
cites work: Uncountable admissibles. II: Compactness
cites work: Σ1 compactness for next admissible sets
cites work: Strict-Π11 predicates on countable and cofinality ω transitive sets
cites work: The fine structure of the constructible hierarchy
cites work: Q3852172
cites work: Generalizing Vaught sentences from ω to strong cofinality ω
cites work: Structures elementarily equivalent relative to infinitary languages to models of higher power
cites work: On the standard part of nonstandard models of set theory
cites work: Uniform inductive definability and infinitary languages
cites work: On partitions into stationary sets
description: scientific article; zbMATH DE number 3889512