About some generalizations of (\(\lambda\) ,\(\mu\) )-compactness. (English) Zbl 0659.03020

The author presents without proof some interesting results concerning the compactness properties of model theoretic logics. He considers the following notion related to (\(\lambda\),\(\mu)\)- and [\(\lambda\),\(\mu\) ]- compactness: A logic L is almost \(\kappa\)-(\(\lambda\),\(\mu)\)-compact, iff for every \(\Gamma\) \(\subseteq L\) of cardinality \(\leq \kappa\), and for every \(\Sigma\) \(\subseteq L\) of cardinality \(\lambda\), it holds: if for every subset \(\Sigma\) ’ of \(\Sigma\) of cardinality \(<\mu\) there is a model of \(\Gamma\) \(\cup \Sigma '\), then for some \(\Sigma^*\subseteq \Sigma\) of cardinality \(\lambda\), \(\Gamma \cup \Sigma^*\) has a model. \(\kappa\)-(\(\lambda\),\(\mu)\)-compactness is the same with the stronger requirement \(\Sigma^*=\Sigma\). Among others, the author states the following results. (a) Every \((\lambda^+,\lambda^+)\)-compact logic is almost \(\lambda^+\)-(\(\lambda\),\(\lambda)\)-compact. (b) If either \(\lambda\) is regular or \(2^{\lambda}=\lambda^+\), then every \((\lambda^+,\lambda^+)\)-compact logic is (\(\lambda\),\(\lambda)\)- compact. (c) Every \(2^{\lambda}\)-(\(\lambda^+,\lambda^+)\)-compact logic is (\(\lambda\),\(\lambda)\)-compact. The paper is connected to the work of J. A. Makowsky and S. Shelah [Ann. Pure Appl. Logic 25, 263-299 (1983; Zbl 0544.03013)] on \([\lambda^+,\lambda^+]\)- compactness and regularity properties of ultrafilters.
Reviewer: J.Oikkonen


03C95 Abstract model theory
03C20 Ultraproducts and related constructions
03E05 Other combinatorial set theory


Zbl 0544.03013