Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-25T05:08:11.649Z Has data issue: false hasContentIssue false
  • English
  • Français

On models with power-like orderings

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah
Saharon Shelah*
Affiliation:
Hebrew University, Jerusalem, Israel Princeton University, Princeton, New Jersey 08540
Get access Rights & Permissions [Opens in a new window]

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
Information
The Journal of Symbolic Logic , Volume 37 , Issue 2 , June 1972 , pp. 247 - 267
Copyright
Copyright © Association for Symbolic Logic 1972

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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. 1148–1155.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. 36–64.Google Scholar
[4]Erdös, P., Hajnal, A. and Rado, R., Partition relations for cardinal numbers, Acta Mathematica, vol. 16 (1965), pp. 93–196.Google Scholar
[5]Frayne, T., Morel, A. and Scott, D., Reduced direct products, Fundamenta Mathematicae, vol. 51 (1962), pp. 195–248.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. 121–131.Google Scholar
[7]Fuhrken, G., Skolem-type normal forms for first-order languages with a generalized quantifier, Fundamenta Mathematicae, vol. 54 (1964), pp. 291–302.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. 249–261.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. 35–62.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. 49–65.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. 265–274.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. 539–543.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. 81–89.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. 390–401.Google Scholar
[19]Barwise, J. and Kunen, K., Hanf numbers for fragments of L∞, ω, Israel Journal of Mathematics, vol. 10 (1971), pp. 306–320.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. 1–94.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. 43–52.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. 257–263.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. 271–362.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. 193–198.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. 50–68.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. 223–238.CrossRefGoogle Scholar
[38]Ebbinghaus, E. D., On models with large automorphism group, Archiv für mathematische Logik und Grundlagenforschung, vol. 14 (1971), pp. 179–197.CrossRefGoogle Scholar
[39]Lipner, L. D., Some aspects of generalized quantifiers Ph.D. thesis, University of California, Berkeley, 1970.Google Scholar