×

\(\omega\)-satisfiability, \(\omega\)-consistency property, and the downward Löwenheim Skolem theorem for \(L_{\kappa,\kappa}\). (English) Zbl 0531.03018

The paper concerns \(L_{\kappa \kappa}\) for strong limit cardinals \(\kappa\) of cofinality \(\omega\). A new version of the notion of a consistency property, that of \(\omega\)-consistency property is introduced. This version has the advantage that sets of sentences in \(\omega\)-consistency properties are \(\omega\)-satisfiable and that every \(\omega\)-satisfiable set of sentences is in an \(\omega\)-consistency property. The paper is connected with earlier work by Karp and Cunningham on chain models. In contrast with earlier versions, Ferro does not need the downward Löwenheim-Skolem Theorem as a tool, but is able to obtain a downward Löwenheim-Skolem Theorem (for \(\omega\)-chains of models) as a corollary to the basic properties of \(\omega\)-consistency properties.
Reviewer: J.Oikkonen

MSC:

03C75 Other infinitary logic
PDFBibTeX XMLCite
Full Text: Numdam EuDML

References:

[1] E. Cunningham , Chain models: applications of consistency properties and back-and-forth techniques in infinite-quantifier languages, Infinitary Logic: in memoriam Carol Karp , Springer-Verlag , Berlin , 1975 . MR 476485
[2] R. Ferro , Consistency property and model existence theorem for second order negative languages with conjunctions and quantifications over sets of cardinality smaller than a strong limit cardinal of denumerable cofinality , Rend. Sem. Mat. Univ. Padova , 55 ( 1976 ), pp. 121 - 141 . Numdam | MR 460065 | Zbl 0365.02006 · Zbl 0365.02006
[3] R. Ferro , An analysis of Karp’s interpolation theorem and the notion of consistency property , Rend. Sem. Univ. Padova , 65 ( 1981 ). Numdam | MR 653287 | Zbl 0485.03014 · Zbl 0485.03014
[4] C.R. Karp , Infinite quantifier languages and \omega -chains of models , Proceedings of the Tarski Symposium, American Mathematical Society , Providence , 1974 . Zbl 0308.02016 · Zbl 0308.02016
[5] H.J. Keisler , Model theory for infinitary logic , North Holland , Amsterdam , 1971 . Zbl 0222.02064 · Zbl 0222.02064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.