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Compactness and Löwenheim-Skolem properties in categories of pre- institutions. (English) Zbl 0792.03026
Rauszer, Cecylia (ed.), Algebraic methods in logic and in computer science. Papers of the XXXVIII semester on algebraic methods in logic and their computer science applications held in Warsaw (Poland) between September 15 and December 15, 1991. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Cent. Publ. 28, 67-94 (1993).
Lindström proved that the compactness and Löwenheim properties characterize first-order logic \(L\) in the following sense: In any logic \(L'\) which strengthens the expressive power of \(L\), one of the properties will fail. In their earlier paper: “A soft stairway to institutions” [Lect. Notes Comput. Sci. 655, 310-329 (1993)], the authors had introduced “pre-institutions” with the intent of providing a general notion of logical system where one could still define interesting morphisms of semantic, syntactic and computational machinery. Building on that paper, the present one deals with arrow-theoretic counterparts of the compactness and Löwenheim properties. The paper culminates with the introduction of cardinal pre-institutions, and the proof that a Löwenheim-Skolem theorem is still valid in such general context.
For the entire collection see [Zbl 0777.00048].
Reviewer: D.Mundici (Milano)

MSC:
03C95 Abstract model theory
68Q65 Abstract data types; algebraic specification
03G30 Categorical logic, topoi
03B10 Classical first-order logic
18B99 Special categories
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