Skip to main content

On the no(M) for M of singular power

Part of the Lecture Notes in Mathematics book series (LNM,volume 1182)

Abstract

We prove that for λ singular of cofinality κ>ℵ0, if (∀μ<λ)μκ then for some model M, M=(M, R M), R a two place predicate, ‖M‖=λ and no (M)={N/≈:N∞, λ M,‖N‖=λ} is quite arbitrary e.g. any μ<λ and λκ (hence 2λ).

See [Sh 5] for the back ground: where the result were proved for M with relations with infinitely many places. By the present paper the only problem left, if we assume V=L, is whether no (M)=λ, may happen for M of cardinality λ for λ singular.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. C.C. Chang, Some remarks on the model theory of infinitary languages:, pp. 36–63 in: The Syntax and Semantics of Infinitary Languages, Lecture Notes in Mathematics, 72, ed., J. Barwise, Springer, 1968.

    Google Scholar 

  2. H.L. Hiller, and S. Shelah, "Singular cohomology in L", Israel Journal of Mathematics. vol. 26 (1977), pp. 313–319.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. M. Nadel and J. Stavi, "L ∞λ-equivalence, isomorphism and potential isomorphism," Transactions of the American Mathematical Society, vol. 236 (1978), pp. 51–74.

    MathSciNet  MATH  Google Scholar 

  4. E. A. Palyutin, "Number of models in L ∞, ω1 theories III," pp. 443–456 in Algebra I Logika, vol. 16, no. 4. (1977): English translation in Algebra and Logic, vol. 16, no. 4 (1977), pp. 299–309.

    CrossRef  MathSciNet  Google Scholar 

  5. D. Scott, "Logic with denumerably long formulas and finite strings of quantifiers," pp. 329–341 in The Theory of Models, North Holland, 1965.

    Google Scholar 

  6. S. Shelah, "On the number of non-isomorphic models of cardinality λ,L ∞λ-equivalent to a fixed model", Notre Dame J. of Formal Logic 22 (1981), 5–10.

    CrossRef  Google Scholar 

  7. —, "On the number of non-isomorphic models of power λ ≡⇔λ to a model of power λ, when λ is weakly compact," Notre Dame J. of Formal Logic, 23 (1982), 21–26.

    CrossRef  Google Scholar 

  8. —, "The consistency of Ext (G, ℤ)=ℚ" Israel Journal of Mathematics, 39 (1981), 283–288.

    CrossRef  MathSciNet  Google Scholar 

  9. —, "A pair of non-isomorphic ≡∞,λ, Models of Power λ for λ singular with λω = λ". Notre Dame J. of Formal Logics, 25 (1984), 97–104.

    CrossRef  Google Scholar 

  10. —, "On the possible number no (M) = the numbers of non isomorphic models L ∞,λ-equivalent of power λ for λ singular, Notre Dame J. Formal Logic, in press.

    Google Scholar 

Download references

Authors

Rights and permissions

Reprints and Permissions

Copyright information

© 1986 Springer-Verlag

About this chapter

Cite this chapter

Shelah, S. (1986). On the no(M) for M of singular power. In: Around Classification Theory of Models. Lecture Notes in Mathematics, vol 1182. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0098507

Download citation

  • DOI: https://doi.org/10.1007/BFb0098507

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16448-7

  • Online ISBN: 978-3-540-39788-5

  • eBook Packages: Springer Book Archive