Skip to main content

Failure of interpolation for quantifiers of monadic type

Part of the Lecture Notes in Mathematics book series (LNM,volume 1130)

Abstract

It is shown that no proper extension of first order logic by Lindström-Mostowski quantifiers of monadic type, that is quantifiers of the form Qx1…xn1(x1),…,фn(xn)), satisfies the many sorted Craig’s interpolation lemma or even the one sorted, if closed under relativizations. For example Lω1ω or any of its admissible fragments can not be generated by any number of these quantifiers. This generalizes previous results of the same type shown under stronger hypothesis. In contrast, all monadic logics generated by cardinal quantifiers satisfy interpolation.

Keywords

  • Extension Property
  • Proper Extension
  • Partial Isomorphism
  • Monadic Predicate
  • Strong Limit Cardinal

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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   54.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   69.95
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. Barwise, K.J. Axioms for abstract model theory. Ann.Math.Logic 7(1974),pp.221–265.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Caicedo,X. Maximality and interpolation in abstract logics. Thesis, University of Maryland (1978)

    Google Scholar 

  3. ____ Banck-and-forth systems for arbitrary quantifiers, IN: "Mathematical Logic in Latin America", Proc. IV Latin American Symposium on Math.Logic, North Holland (1980).

    Google Scholar 

  4. ____ On extensions of Lωω(Q1), Notre Dame J. of Formal Logic 22 (1981),pp. 85–93.

    CrossRef  MathSciNet  Google Scholar 

  5. Fajardo, S. Compacidad y decidibilidad en lógicas monádicas con cuantificadores cardinales, Rev.Colombiana de Mat. 14(1980), pp. 173–196.

    MathSciNet  Google Scholar 

  6. Flum,J. Characterizing logics. Preprint (1982).

    Google Scholar 

  7. Friedman, H. Beth’s theorem in cardinality logics. Israel J. Math. 14(1973),pp.205–212.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. Lindstrom, P. First order predicate calculus with generalized quantifiers. Theoria 32(1966), pp. 187–195.

    MATH  Google Scholar 

  9. ____ On extensions of elementary logic, Theoria 35 (1969), pp. 1–11.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Makowsky, J.A. Characterizing monadic and equivalence quantifiers, Preprint (1978)

    Google Scholar 

  11. Makowsky, J.A. and Shelah, S. The theorems of Beth and Craig in abstract model theory I, Trans.AMS 256(1979) pp.215–239.

    MathSciNet  MATH  Google Scholar 

  12. Mostowski, A. On a generalization of quantifiers, Fund.Math 44(1957),pp.12–36.

    MathSciNet  MATH  Google Scholar 

  13. Mundici, D. Quantifiers, an owerview. Preprint (1981).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1985 Springer-Verlag

About this paper

Cite this paper

Caicedo, X. (1985). Failure of interpolation for quantifiers of monadic type. In: Di Prisco, C.A. (eds) Methods in Mathematical Logic. Lecture Notes in Mathematics, vol 1130. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0075304

Download citation

  • DOI: https://doi.org/10.1007/BFb0075304

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-15236-1

  • Online ISBN: 978-3-540-39414-3

  • eBook Packages: Springer Book Archive