Remarks in abstract model theory
type: Item
instance of: scholarly article
title: Remarks in abstract model theory
zbMATH Open document ID: 0599.03037
DOI: 10.1016/0168-0072(85)90002-8
author: S. Shelah
published in: Annals of Pure and Applied Logic
publication date: 1985-01-01T00:00:00Z
review text: This rather long paper consists of two distinct parts, the first of which is about the Beth definability property, and the second about compactness. The first half of Part I contains abstract results about Beth closures and preservation properties. At the beginning, a one-step Beth closure \(L^ B\) is defined by adding new formulas for the required explicit definitions. Then the Beth closure \(L^{Beth}\) is defined by iteration. A sufficient condition is given to the compactness of the one-step Beth closure. Then the pair preservation property PPP is related to the reduction property \(URP_ 2\) and the preservation of these under Beth closures is studied. The first half of Part I ends with a proof of the consistency of that there is no definable extension of \(L(Q_ 1)\) with PPP and interpolation. The latter half of Part I contains results about cofinality quantifiers and stationary logic. The quantifier \(Q_ C^{cf}\) says that the cofinality belongs to C. \(L(aa_{\lambda})\) is the analogue of L(aa) for \(\lambda\). The first result is that the pair \((L(Q^{cf}_{\aleph_ 0},L(aa))\) satisfies the interpolation theorem. It is shown that the pair \((L(Q^{cf}_{\leq \lambda},L(aa_{\lambda}))\) has the super \(\aleph_ 0\)-homogeneity property. It says the following. Assume that in a model M to every \(L(aa_{\lambda})\)-formula \(\psi\) corresponds a relation symbol \(R_{\psi}\) whose interpretation is the relation defined by \(\psi\). Then M is \(L(Q^{cf}_{\leq \lambda})\)-equivalent to a model N where any two finite sequences satisfying the same \(L(Q^{cf}_{\leq \lambda})\)- formulas can be mapped onto each other by an automorphism of N. The (full) Beth closure of \(L(Q^{cf}_{\leq \kappa})\) \((\kappa \geq 2^{\aleph_ 0})\) has the following properties: (1) it is compact, (2) the pair \((L(Q^{cf}_{\leq \kappa}),L(aa_{\kappa}))\) has the interpolation and super \(\aleph_ 0\)-homogeneity properties, (3) the Beth closure of \(L(Q^{cf}_{\leq \kappa})\) has (trivially) the Be…
reviewed by: J.Oikkonen
Mathematics Subject Classification ID: 03C95
Mathematics Subject Classification ID: 03C40
Mathematics Subject Classification ID: 03C80
Mathematics Subject Classification ID: 03C55
zbMATH DE Number: 3966047
zbMATH Keywords: abstract logic
zbMATH Keywords: Beth definability property
zbMATH Keywords: compactness
zbMATH Keywords: Beth closures
zbMATH Keywords: pair preservation property
zbMATH Keywords: cofinality quantifiers
zbMATH Keywords: stationary logic
zbMATH Keywords: interpolation
zbMATH Keywords: occurrence numbers
zbMATH Keywords: amalgamation
MaRDI profile type: Publication
full work available at URL: 0168-0072(85)90002-8
OpenAlex ID: W2002354590
cites work: A correction to “stationary logic”
cites work: Stationary logic of finitely determinate structures
cites work: The Hanf number of stationary logic
cites work: Q3917485
cites work: Two notes on abstract model theory. I. Properties invariant on the range of definable relations between structures
cites work: Q4068717
cites work: Two notes on abstract model theory. II. Languages for which, the set of valid sentences is semi-invariantly implicitly definable
cites work: Beth's theorem in cardinality logics
cites work: Theorems of Beth and Craig in Abstract Model Theory. I The Abstract Settin
cites work: The theorems of beth and Craig in abstract model theory II. Compact logics
cites work: Positive results in abstract model theory: a theory of compact logics
cites work: δ-Logics and generalized quantifiers
cites work: Stationary logic and its friends. I
cites work: Stationary logic and its friends. II
cites work: Two cardinal compactness
cites work: On models with power-like orderings
cites work: Generalized Quantifiers and Compact Logic
cites work: A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals
cites work: Decomposing uncountable squares to countably many chains
cites work: Models with second order properties I. Boolean algebras with no definable automorphisms
cites work: Models with second order properties II. Trees with no undefined branches
cites work: Classification theory and the number of non-isomorphic models
cites work: Models with second order properties. IV. A general method and eliminating diamonds
cites work: More on powers of singular cardinals
cites work: Proper forcing
Wikidata QID: Q127441804
description: scientific article; zbMATH DE number 3966047