Hostname: page-component-76fb5796d-r6qrq Total loading time: 0 Render date: 2024-04-25T12:52:35.067Z Has data issue: false hasContentIssue false
  • English
  • Français

Peano arithmetic may not be interpretable in the monadic theory of linear orders

Published online by Cambridge University Press:  12 March 2014

Shmuel Lifsches  and
Saharon Shelah
Shmuel Lifsches
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem, Israel, E-mail: sam@math.huji.ac.il
Saharon Shelah
Affiliation:
Institute of Mathematics, The Hebrew University, Jerusalem, Israel, E-mail: shelah@math.huji.ac.il
Get access Rights & Permissions [Opens in a new window]

Abstract

Gurevich and Shelah have shown that Peano Arithmetic cannot be interpreted in the monadic second-order theory of short chains (hence, in the monadic second-order theory of the real line). We will show here that it is consistent that the monadic second-order theory of no chain interprets Peano Arithmetic.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 62 , Issue 3 , September 1997 , pp. 848 - 872
Copyright
Copyright © Association for Symbolic Logic 1997

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]Baldwin, J., Definable second-order quantifiers, Model theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, Berlin, 1985, pp. 445–477.Google Scholar
[2]Baldwin, J. and Shelah, S., Classification of theories by second order quantifiers, Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 229–303.CrossRefGoogle Scholar
[3]Gurevich, Y., Monadic second-order theories, (Barwise, J. and Feferman, S., editors), Springer-Verlag, Berlin, 1985, pp. 479–506.Google Scholar
[4]Gurevich, Y., Magidor, M., and Shelah, S., The monadic theory of ω 2, this Journal, vol. 48 (1983), pp. 387–398.Google Scholar
[5]Gurevich, Y., Magidor, M., and Shelah, S., On the strength of the interpretation method, this Journal, vol. 54 (1989), pp. 305–323.Google Scholar
[6]Gurevich, Y. and Shelah, S., Monadic theory of order and topology in ZFC, Annals of Mathematical Logic, vol. 23 (1982), pp. 179–182.CrossRefGoogle Scholar
[7]Gurevich, Y. and Shelah, S., Interpreting second-order logic in the monadic theory of order, this Journal, vol. 48 (1983), pp. 816–828.Google Scholar
[8]Gurevich, Y. and Shelah, S., The monadic theory and the ‘next world’, Israel Journal of Mathematics, vol. 49 (1984), pp. 55–68.CrossRefGoogle Scholar
[9]Lifsches, S. and Shelah, S., Random graphs in the monadic theory of order, in preparation.Google Scholar
[10]Shelah, S., The monadic theory of order, Annals of Mathematics, vol. 102 (1975), pp. 379–419.CrossRefGoogle Scholar
[11]Shelah, S., Notes on monadic logic, Part B: Complexity of linear orders, Israel Journal of Mathematics, vol. 69 (1990), pp. 99–116.CrossRefGoogle Scholar
[12]Tarski, A., Mostowski, A., and Robinson, R. M., Undecidable theories, North-Holland Publishing Company, 1953, pp. xi and 98.Google Scholar