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On models with power-like orderings

Published online by Cambridge University Press:  12 March 2014

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

Abstract

We prove here theorems of the form: if T has a model M in which P1(M) is κ1-like ordered, P2(M) is κ2-like ordered …, and Q1(M) is of power λ1, …, then T has a model N in which P1(M) is κ1-like ordered …, Q1(N) is of power λ1, …. (In this article κ is a strong-limit singular cardinal, and κ′ is a singular cardinal.)

We also sometimes add the condition that M, N omits some types. The results are seemingly the best possible, i.e. according to our knowledge about n-cardinal problems (or, more precisely, a certain variant of them).

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1972

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References

REFERENCES

[1]Chang, C. C., Two refinements of Morley's method on omitting types of elements, Notices of the American Mathematical Society, vol. 11 (1964), p. 679.Google Scholar
[2]Chang, C. C., A note on a two cardinal problem, Proceedings of the American Mathematical Society, vol. 16 (1965), pp. 11481155.CrossRefGoogle Scholar
[3]Chang, C. C., Some remarks on the model theory of infinitary languages, Lecture notes in mathematics, no. 72, Springer-Verlag, New York, 1968, pp. 3664.Google Scholar
[4]Erdös, P., Hajnal, A. and Rado, R., Partition relations for cardinal numbers, Acta Mathematica, vol. 16 (1965), pp. 93196.Google Scholar
[5]Frayne, T., Morel, A. and Scott, D., Reduced direct products, Fundamenta Mathematicae, vol. 51 (1962), pp. 195248.CrossRefGoogle Scholar
[6]Fuhrken, G., Languages with the added quantifier “there exists at least ℵα, The theory of models (Addison, J. W., Henkin, L. and Tarski, A., Editors), North-Holland, Amsterdam, 1965, pp. 121131.Google Scholar
[7]Fuhrken, G., Skolem-type normal forms for first-order languages with a generalized quantifier, Fundamenta Mathematicae, vol. 54 (1964), pp. 291302.CrossRefGoogle Scholar
[8]Helling, M., Hanf numbers for some generalizations of first-order languages, Notices of the American Mathematical Society, vol. 11 (1964), p. 679.Google Scholar
[9]Keisler, H. J., Some model theoretic results for ω-logic, Israel Journal of Mathematics, vol. 4 (1966), pp. 249261.CrossRefGoogle Scholar
[10]Keisler, H. J., Models with ordering, Logic, methodology and philosophy of science. III (Van Rootselaar, B. and Stoal, J. F., Editors), North-Holland, Amsterdam, 1968, pp. 3562.CrossRefGoogle Scholar
[11]Keisler, H. J., Weakly well-ordered models, Notices of the American Mathematical Society, vol. 14 (1967), p. 414.Google Scholar
[12]Keisler, H. J. and Morley, M., Elementary extensions of models of set theory, Israel Journal of Mathematics, vol. 6 (1968), pp. 4965.CrossRefGoogle Scholar
[13]Morley, M., Omitting classes of elements, The theory of models (Addison, J. W., Henkin, L. and Tarski, A., Editors), North-Holland, Amsterdam, 1965, pp. 265274.Google Scholar
[14]Morley, M. and Morley, V., The Hanf number for κ-logic, Notices of the American Mathematical Society, vol. 14 (1967), p. 556.Google Scholar
[15]Shelah, S., On models with orderings, Notices of the American Mathematical Society, vol. 16 (1969), p. 580.Google Scholar
[16]Shelah, S., A note on Hanf numbers, Pacific Journal of Mathematics, vol. 16 (1970), pp. 539543.Google Scholar
[17]Vaught, R. L., The Löwenheim-Skolem theorem, Logic, methodology and philosophy of science (Bar-Hillel, Y., Editor), North-Holland, Amsterdam, 1965, pp. 8189.Google Scholar
[18]Vaught, R. L., A Löwenheim-Skolem theorem for cardinals far apart, The theory of models (Addison, J. W., Henkin, L. and Tarski, A., Editors), North-Holland, Amsterdam, 1965, pp. 390401.Google Scholar
[19]Barwise, J. and Kunen, K., Hanf numbers for fragments of L∞, ω, Israel Journal of Mathematics, vol. 10 (1971), pp. 306320.CrossRefGoogle Scholar
[20]Friedman, H., Back and forth theorem L(Q), L∞, ω(Q) and Beth's theorem, Stanford University, Stanford, Calif., 1971 (mimeographed notes).Google Scholar
[21]Keisler, H. J., Model theory for infinitary logic, North-Holland, Amsterdam, 1971.Google Scholar
[22]Keisler, H. J., Logic with the added quantifier, “there exists uncountably many”, Annals of Mathematical Logic, vol. 1 (1970), pp. 194.CrossRefGoogle Scholar
[23]Morley, M. D., The Löwenheim-Skolem theorem for models with standard part, Symposia Mathematica, vol. 5, Academic Press, London and New York, 1971, pp. 4352.Google Scholar
[24]MacDowell, R. and Specker, E., Modelle der Arithmetik, Infinitistic methods (Proceedings of the Symposium on Foundation of Mathematics (Warsaw, 1959)), New York, 1961, pp. 257263.Google Scholar
[25]Schmerl, J. H., On χ-like models for inaccessible χ, Ph.D. thesis, University of California, Berkeley, Calif., 1971.Google Scholar
[26]Schmerl, J. H., An elementary sentence which has ordered models (to appear).Google Scholar
[27]Schmerl, J. H. and Shelah, S., On models with orderings, Notices of the American Mathematical Society, vol. 16 (1969), p. 840.Google Scholar
[28]Schmerl, J. H. and Shelah, S., On power-like models for hyperinaccessible cardinals (to appear).Google Scholar
[29]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. 271362.CrossRefGoogle Scholar
[30]Shelah, S., Categoricity of classes of models, Ph.D. thesis, Hebrew University, Jerusalem, Israel, 1969.Google Scholar
[31]Shelah, S., On generalizations of categoricity, Notices of the American Mathematical Society, vol. 16 (1969), p. 683.Google Scholar
[32]Shelah, S., Two-cardinal compactness, Israel Journal of Mathematics, vol. 7 (1971), pp. 193198.CrossRefGoogle Scholar
[33]Shelah, S., Two-cardinal and power-like models: compactness and large group of automorphisms, Notices of the American Mathematical Society, vol. 18 (1971); p. 425.Google Scholar
[34]Viner, S., Some problems in first-order predicate calculus with numerical quantifiers, Ph.D. thesis, Hebrew University, Jerusalem, Israel, 1971; Notices of the American Mathematical Society, vol. 17 (1970), pp. 456, 964, 1077.Google Scholar
[35]Ehrenfeucht, A. and Mostowski, A., Models of axiomatic theories admitting automorphisms, Fundamenta Mathematicae, vol. 43 (1956), pp. 5068.CrossRefGoogle Scholar
[36]Ehrenfeucht, A., Theories having at least continuum many nonisomorphic models in each infinite power, Notices of the American Mathematical Society, vol. 5 (1968), p. 680.Google Scholar
[37]Levy, A., Axiom schemata of strong infinity in axiomatic set theory, Pacific Journal of Mathematics, vol. 10 (1960), pp. 223238.CrossRefGoogle Scholar
[38]Ebbinghaus, E. D., On models with large automorphism group, Archiv für mathematische Logik und Grundlagenforschung, vol. 14 (1971), pp. 179197.CrossRefGoogle Scholar
[39]Lipner, L. D., Some aspects of generalized quantifiers Ph.D. thesis, University of California, Berkeley, 1970.Google Scholar