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Interpolation, compactness and JEP in soft model theory

Interpolation, compactness and JEP in soft model theory Let[Figure not available: see fulltext.] be the following statement: “for any infinite regularκ, for any uniform ultrafilterD onκ,D isλ-descendingly incomplete for all infiniteλ”.[Figure not available: see fulltext.] is weaker than ⌍0#. Assuming[Figure not available: see fulltext.] we prove the following: letL be a logic in which the class of sentences of typeτ is a set if so isτ; then: (I)L is compact iffL has JEP; (II)L satisfies Robinson Consistency Theorem iffL is compact and satisfies Craig Interpolation theorem; (III) if, in addition,L is single-sorted, thenL satisfies Robinson Consistency Theorem iffL has JEP#. JEP (resp. JEP#) are the natural generalizations for logicL of the familiar Joint Embedding Property of elementary (resp. complete) embeddings in first order logic. As a corollary, we characterize first order logic as the only logic having Löwenheim number equal toω together with JEP. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Mathematical Logic Springer Journals

Interpolation, compactness and JEP in soft model theory

Mundici, Daniele
Archive for Mathematical Logic , Volume 22 (2) – Jan 20, 2006
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References (15)

Publisher
Springer Journals
Copyright
Copyright © 1980 by Verlag W. Kohlhammer
Subject
Mathematics; Mathematical Logic and Foundations; Mathematics, general; Algebra
ISSN
0933-5846
eISSN
1432-0665
DOI
10.1007/BF02318027
Publisher site
See Article on Publisher Site

Abstract

Let[Figure not available: see fulltext.] be the following statement: “for any infinite regularκ, for any uniform ultrafilterD onκ,D isλ-descendingly incomplete for all infiniteλ”.[Figure not available: see fulltext.] is weaker than ⌍0#. Assuming[Figure not available: see fulltext.] we prove the following: letL be a logic in which the class of sentences of typeτ is a set if so isτ; then: (I)L is compact iffL has JEP; (II)L satisfies Robinson Consistency Theorem iffL is compact and satisfies Craig Interpolation theorem; (III) if, in addition,L is single-sorted, thenL satisfies Robinson Consistency Theorem iffL has JEP#. JEP (resp. JEP#) are the natural generalizations for logicL of the familiar Joint Embedding Property of elementary (resp. complete) embeddings in first order logic. As a corollary, we characterize first order logic as the only logic having Löwenheim number equal toω together with JEP.

Journal

Archive for Mathematical Logic – Springer Journals

Published: Jan 20, 2006

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