``Gap 1'' two-cardinal principles and the omitting types theorem for \(\mathcal L(\mathcal Q)\)
type: Item
instance of: scholarly article
title: ``Gap 1'' two-cardinal principles and the omitting types theorem for \(\mathcal L(\mathcal Q)\)
zbMATH Open document ID: 0681.03016
DOI: 10.1007/BF02764857
author: S. Shelah
published in: Israel Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: The author states two goals: (1) to give precise set-theoretic equivalents to two-cardinal transfer principles for strong limit singular cardinals, (2) to settle the relationship of two-cardinal transfer principles to omitting types theorems for \({\mathcal L}(Q)\). The main result is the expansion of the following. For \(\lambda\) a singular strong limit cardinal, and more generally if \(\lambda >2^{cof \lambda}\), the following are equivalent: (1) \((\aleph_ 0,\aleph_ 1)\to (\lambda,\lambda^+),\) (2) Completeness theorem for \({\mathcal L}(Q)\) in the \(\lambda^+\)- interpretation, (3) Omitting types for \({\mathcal L}(Q)\) in the \(\lambda^+\)- interpretation, (4) Various weak forms of \(\square_{\lambda}.\) The latter part of the paper is concerned with how to carry out a Henkin construction for \({\mathcal L}(Q)\) over a tree of approximations to a model of size \(\lambda^+\), given a suitable \(\square\)-like principle.
Mathematics Subject Classification ID: 03C55
Mathematics Subject Classification ID: 03E05
zbMATH DE Number: 4114600
zbMATH Keywords: set-theoretic equivalents to two-cardinal transfer principles
zbMATH Keywords: strong limit singular cardinals
zbMATH Keywords: omitting types theorems
zbMATH Keywords: Henkin construction
reviewed by: J. M. Plotkin
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02764857
OpenAlex ID: W2056201127
description: scientific article; zbMATH DE number 4114600