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Infinitary formulas preserved under unions of models1

  • Bienvenido F. Nebres (a1)

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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.

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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.

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[1]Barwise, J., Infinitary logic and admissible sets, this Journal, vol. 34 (1969), pp. 226252.
[2]Barwise, J., Remarks on universal sentences of , Duke Mathematical Journal, vol. 36 (1969), pp. 631637.
[3]Barwise, J., Infinitary logic and admissible sets, Dissertation, Stanford University, 1967.
[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. 309342.
[5]Chang, C. C., On unions of chains of models, Proceedings of the American Mathematical Society, vol. 10 (1959), pp. 120127.
[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. 1107.
[7]Feferman, S., Persistent and invariant formulas for outer extensions, Compositio Mathematica, vol. 20 (1969), pp. 2952.
[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. 480485.
[9]Henkin, L., An extension of the Craig-Lyndon interpolation theorem, this Journal, vol. 28 (1963), pp. 201206.
[10]Hintikka, J., Form and content in quantification theory, Acta Philosophica Fennica, vol. 8 (1955), pp. 755.
[11]Kanger, S., Provability in logic, Almquist and Wiksell, Stockholm, 1957.
[12]Karp, C., Languages with expressions of infinite length, North-Holland, Amsterdam, 1964.
[13]Keisler, H. J., Model theory for , Unpublished Lecture Notes, University of Wisconsin, 1969.
[14]Keisler, H. J., Some applications of infinitely long formulas, this Journal, vol. 30 (1965), pp. 339349.
[15]Keisler, H. J., Unions of relational systems, Proceedings of the American Mathematical Society, vol. 15 (1964), pp. 540545.
[16]Lopez-Escobar, E. G. K., An interpolation theorem for denumerably long formulas, Fundamenta Mathematicae, vol. 57 (1965), pp. 254272.
[17]Makkai, M., On the model theory of denumerably long formulas with finite strings of quantifiers, this Journal, vol. 34 (1969), pp. 437459.
[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. 341346.
[19]Malitz, J., Problems in the model theory of infinitary languges, Doctoral Dissertation, University of California, Berkeley, 1965.
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[25]Weinstein, J., 1, ω) Properties of unions of models, The syntax and semantics of infinitary languages, Lecture Notes in Mathematics, No. 72, Springer-Verlag, 1968, pp. 265268.

Infinitary formulas preserved under unions of models1

  • Bienvenido F. Nebres (a1)

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