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Weak definability in infinitary languages

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah
Saharon Shelah*
Affiliation:
Princeton University, Princeton, New Jersey 08540 Hebrew University, Jerusalem, Israel
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Abstract

We shall prove that if a model of cardinality κ can be expanded to a model of a sentence ψ of by adding a suitable predicate in more than κ ways, then, it has a submodel of power μ which can be expanded to a model of ψ in > μ ways provided that λ, κ, μ satisfy suitable conditions.

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Type
Research Article
Information
The Journal of Symbolic Logic , Volume 38 , Issue 3 , September 1973 , pp. 399 - 404
Copyright
Copyright © Association for Symbolic Logic 1973

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References

REFERENCES

[1]Baumgartner, J. E., Results and independence proofs in combinatorial set-theory, Ph.D. thesis, University of California, Berkeley, 1970.Google Scholar
[2]Baumgartner, J. E., On the cardinality of dense subsets of linear orderings. I, Notices of the American Mathematical Society, vol. 15 (1968), p. 935. Abstract #68T-E33.Google Scholar
[3]Beth, E. W., On Padoa's method in the theory of definitions, Indagationes Mathematicae, vol. 15 (1953), pp. 330–339.CrossRefGoogle Scholar
[4]Chang, C. C., Some new results in definability, Bulletin of the American Mathematical Society, vol. 70 (1964), pp. 808–813.CrossRefGoogle Scholar
[5]Friedman, H., Back and forth, L(Q), L∞, ω(Q) and Beth theorem, mimeograph, Stanford University, 11 1971; Israel Journal of Mathematics (to appear).Google Scholar
[6]Keisler, H. J., Model theory for infinitary logic, North-Holland, Amsterdam, 1971.Google Scholar
[7]Kueker, D., Definability, automorphisms and infinitary languages, the syntax and semantics of infinitary languages (Barwise, J., Editor), Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1972, pp. 152–165.Google Scholar
[8]Kunen, K., Implicit definability and infinitary language, this Journal, vol. 33 (1968), pp. 446–451.Google Scholar
[9]Makkai, M., A generalization of a theorem of E. W. Beth, Acta Mathematica Academiae Scientiarum Hungaricae, vol. 15 (1964), p. 227.CrossRefGoogle Scholar
[10]Malitz, J., Infinitary analogs of theorems from first order model theory, this Journal, vol. 36 (1971), pp. 216–228.Google Scholar
[11]Prikry, K., Changing measurable into accessible cardinals, Dissertationes Mathematicae Rozprawy Matematyczne, No. 68, Warsaw, 1970.Google Scholar
[12]Reyes, G. E., Local definability theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 95–137.CrossRefGoogle Scholar
[13]Shelah, S., Generalizations of saturativity, Notices of the American Mathematical Society, vol. 18 (1971), p. 258. Abstract #71T-E2.Google Scholar
[14]Shelah, S., The number of nonisomorphic models of an unstable first order theory, Israel Journal of Mathematics, vol. 9 (1971), pp. 473–487.CrossRefGoogle Scholar
[15]Shelah, S., Notes in combinatorial set theory, Israel Journal of Mathematics (to appear).Google Scholar
[16]Shelah, S., Remark to “Local definability theory” of Reyes, Annals of Mathematical Logic, vol. 2 (1971), pp. 441–447.Google Scholar
[17]Shelah, S., Weak definability for infinitary languages, Notices of the American Mathematical Society, vol. 17 (1970), p. 834. Abstract #70T–E57.Google Scholar
[18]Silver, J. (to appear).Google Scholar
[19]Lopez-Escobar, E., An interpolation theory for denumerably long sentences, Fundamenta Mathematicae, vol. 57 (1965), pp. 253–272.Google Scholar
[20]Svenonius, L., A theorem on permutations in models, Theoria, vol. 25 (1959), pp. 173–178.CrossRefGoogle Scholar
[21]Shelah, S., Stability, the f.c.p. and superstability, model-theoretic properties of formulas in first order theory, Annals of Mathematical Logic, vol. 3 (1971), pp. 262–271.CrossRefGoogle Scholar
[22]Kueker, D. W., Generalized interpolation and definability, Annals of Mathematical Logic, vol. 1 (1970), pp. 423–468.Google Scholar