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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

BIBLIOGRAPHY

[1]Barwise, J., Infinitary logic and admissible sets, this Journal, vol. 34 (1969), pp. 226–252.Google Scholar
[2]Barwise, J., Remarks on universal sentences of , Duke Mathematical Journal, vol. 36 (1969), pp. 631–637.CrossRefGoogle Scholar
[3]Barwise, J., Infinitary logic and admissible sets, Dissertation, Stanford University, 1967.Google Scholar
[4]Beth, E., Semantic entailment and formal derivability, Mededelingen van de Koninklijke Vlaamse Academic voor Wetenschappen, Letteren en Schone Kunsten van Belgie, vol. 18 (1955), pp. 309–342.Google Scholar
[5]Chang, C. C., On unions of chains of models, Proceedings of the American Mathematical Society, vol. 10 (1959), pp. 120–127.CrossRefGoogle Scholar
[6]Feferman, S., Lectures on proof theory, Proceedings of the Summer School in Logic (Leeds, 1967), Lecture Notes in Mathematics, no. 70, Springer-Verlag, 1968, pp. 1–107.Google Scholar
[7]Feferman, S., Persistent and invariant formulas for outer extensions, Compositio Mathematica, vol. 20 (1969), pp. 29–52.Google Scholar
[8]Feferman, S. and Kreisel, G., Persistent and invariant formulas relative to theories of higher order, Bulletin of the American Mathematical Society, vol. 72 (1966), pp. 480–485.CrossRefGoogle Scholar
[9]Henkin, L., An extension of the Craig-Lyndon interpolation theorem, this Journal, vol. 28 (1963), pp. 201–206.Google Scholar
[10]Hintikka, J., Form and content in quantification theory, Acta Philosophica Fennica, vol. 8 (1955), pp. 7–55.Google Scholar
[11]Kanger, S., Provability in logic, Almquist and Wiksell, Stockholm, 1957.Google Scholar
[12]Karp, C., Languages with expressions of infinite length, North-Holland, Amsterdam, 1964.Google Scholar
[13]Keisler, H. J., Model theory for , Unpublished Lecture Notes, University of Wisconsin, 1969.Google Scholar
[14]Keisler, H. J., Some applications of infinitely long formulas, this Journal, vol. 30 (1965), pp. 339–349.Google Scholar
[15]Keisler, H. J., Unions of relational systems, Proceedings of the American Mathematical Society, vol. 15 (1964), pp. 540–545.CrossRefGoogle Scholar
[16]Lopez-Escobar, E. G. K., An interpolation theorem for denumerably long formulas, Fundamenta Mathematicae, vol. 57 (1965), pp. 254–272.CrossRefGoogle Scholar
[17]Makkai, M., On the model theory of denumerably long formulas with finite strings of quantifiers, this Journal, vol. 34 (1969), pp. 437–459.Google Scholar
[18]Makkai, M., An application of a method of Smullyan's to logic on admissible sets, Bulletin de l'Académie Polonaise des Sciences, vol. 17 (1969), pp. 341–346.Google Scholar
[19]Malitz, J., Problems in the model theory of infinitary languges, Doctoral Dissertation, University of California, Berkeley, 1965.Google Scholar
[20]Malitz, J., Universal classes in infinitary languages, Duke Mathematical Journal, vol. 36 (1969), pp. 621–630.CrossRefGoogle Scholar
[21]Nebres, B., A syntactic characterization of infinitary sentences preserved under unions of models, Notices of the American Mathematical Society, vol. 16 (1969), pp. 423–424.Google Scholar
[22]Schütte, K., Ein System des verknüpfenden Schliessens, Archiv für mathematische Logik und Grundlagenforschung, vol. 2 (1956), pp. 56–57.Google Scholar
[23]Scott, D., Logic with denumerably long formulas and finite strings of quantifiers, The theory of models, North-Holland, Amsterdam, 1965, pp. 329–341.Google Scholar
[24]Smullyan, R., First-order logic, Springer-Verlag, New York, 1968.CrossRefGoogle Scholar
[25]Weinstein, J., (ω1, ω) Properties of unions of models, The syntax and semantics of infinitary languages, Lecture Notes in Mathematics, No. 72, Springer-Verlag, 1968, pp. 265–268.Google Scholar