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Infinitary formulas preserved under unions of models1

Published online by Cambridge University Press:  12 March 2014

Bienvenido F. Nebres
Bienvenido F. Nebres*
Affiliation:
Ateneo De Manila University, Manila, Philippines
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Extract

So-called “preservation theorems” relate the (possible) syntactic form of the axioms of a theory to certain closure conditions on its class of models. Such results are well known for the first-order predicate calculus, Lω, ω, and there are various expositions; e.g., Keisler [14], [15]. For the language , the first results were the theorems of Lopez-Escobar on sentences preserved under homomorphic images and of Malitz on formulas preserved under substructures. More recently, Feferman added a result on formulas preserved under (or persistent for) ∈-extensions. Some of these theorems will be considered in subsequent sections. A more thorough treatment may be found in Makkai [17]. The main new preservation result obtained here characterizes the sentences preserved under ω-unions. This notion and the statement of the theorem will be explained shortly.

It is a familiar experience in mathematical research that concepts which are equivalent in a special case diverge in general. In the case at hand, one must expect to consider different possible statements for , which generalize a known result for Lω, ω. Moreover, diverse proofs may yield the same result in the special case, not all of which can be extended to the general case. Again, since the compactness theorem fails for , one cannot expect to extend the arguments from Lω, ω which use this in an essential way.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 37 , Issue 3 , September 1972 , pp. 449 - 465
Copyright
Copyright © Association for Symbolic Logic 1972

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Footnotes

1

This paper forms part of the author's Ph.D. thesis, submitted to Stanford University in May, 1970. We would like to thank our thesis adviser, Professor Solomon Feferman, for his advice and direction and unfailing willingness to give of his time and effort throughout our work. We also thank Professor Georg Kreisel for his interest and many helpful suggestions.

References

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