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Infinitary analogs of theorems from first order model theory

Published online by Cambridge University Press:  12 March 2014

Jerome Malitz
Jerome Malitz*
Affiliation:
University of Colorado, Boulder, Colorado 80302
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Extract

The material presented here belongs to the model theory of the Lκ, λ languages. Our results are either infinitary analogs of important theorems in finitary model theory, or else show that such analogs do not exist.

For example, it is well known that whenever i, and , have the same true Lω, ω sentences (i.e., are elementarily equivalent) for i = 1, 2, then the cardinal sums 1 + 2 and + have the same true Lω, ω sentences, and the direct products 1 · 2 and · have the same true Lω, ω sentences [3]. We show that this is true when ‘Lω, ω’ is replaced by ‘Lκ, λ’ if and only if κ is strongly inaccessible. For Lω1, ω this settles a question, posed by Lopez-Escobar [7].

In §3 we give a complete description of the expressive power of those sentences of Lκ, λ in which the identity symbol is the only relation symbol which occurs. This extends a result by Hanf [4].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 2 , June 1971 , pp. 216 - 228
Copyright
Copyright © Association for Symbolic Logic 1971

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References

[1]Beth, E., On Padua's method in the theory of definition, Proceedings of the Royal Society, Series A, vol. 56 (1953), pp. 330–339.Google Scholar
[2]Ehrenfeucht, A. and Mostowski, A., Models of axiomatic theories admitting automorphisms, Fundamenta Matkematicae, vol. 43 (1956), pp. 40–68.Google Scholar
[3]Feferman, and Vaught, , The first order properties of algebraic systems, Fundamenta Mathematicae, vol. 47 (1959), pp. 57–103.CrossRefGoogle Scholar
[4]Hanf, W., Fundamental problems concerning languages with infinitely long expressions, Doctoral Dissertation.Google Scholar
[5]Henkin, L., An extension of the Craig-Lyndon interpolation theorem, this Journal, vol. 28 (1963), pp. 201–216.Google Scholar
[6]Karp, C., Languages with expressions of infinite length, North-Holland, Amsterdam, 1964.Google Scholar
[7]Lopez-Escobar, E. G. K., Infinitely long formulas with countable quantifier degrees, Doctoral Dissertation, Berkeley, California, 1964.Google Scholar
[8]Lopez-Escobar, E. G. K., An interpolation theorem for denumerably long formulas, Fundamenta Matkematicae, vol. 57 (1965).Google Scholar
[9]Lopez-Escobar, E. G. K., On defining well-orderings, Fundamenta Matkematicae, vol. 59 (1966).Google Scholar
[10]Lopez-Escobar, E. G. K., An addition to “On defining well-orderings”, Fundamenta Matkematicae, vol. 59 (1966).Google Scholar
[11]Maehara, S. and Takeuti, G., A formal system of first order predicate calculus with infinitely long expressions, Journal of the Mathematical Society of Japan, vol. 13 (1961), pp. 357–370.CrossRefGoogle Scholar
[12]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
[13]Malitz, J., Cardinal sums and Beth's theorem in infinitary languages, Notices of the American Mathematical Society, vol. 12 (1965), no. 3, p. 379.Google Scholar
[14]Malitz, J., The failure of Craig's theorem in languages with uncountable conjunctions, Notices of the American Mathematical Society, vol. 12 (1965), no. 4, p. 467.Google Scholar
[15]Morley, M., Omitting classes of elements, The theory of models, Proceedings of the 1963 International Symposium at Berkeley, North-Holland, Amsterdam, 1965, pp. 268–323.Google Scholar
[16]Scott, D., Logic with denumerably long formulas and finite strings of quantifiers, The theory of models, Proceedings of the 1963 International Symposium at Berkeley, North-Holland, Amsterdam, 1965, pp. 329–341.Google Scholar