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

  • Shmuel Lifsches (a1) and Saharon Shelah (a2)

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.

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[1]Baldwin, J., Definable second-order quantifiers, Model theoretic logics (Barwise, J. and Feferman, S., editors), Springer-Verlag, Berlin, 1985, pp. 445477.
[2]Baldwin, J. and Shelah, S., Classification of theories by second order quantifiers, Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 229303.
[3]Gurevich, Y., Monadic second-order theories, (Barwise, J. and Feferman, S., editors), Springer-Verlag, Berlin, 1985, pp. 479506.
[4]Gurevich, Y., Magidor, M., and Shelah, S., The monadic theory of ω 2, this Journal, vol. 48 (1983), pp. 387398.
[5]Gurevich, Y., Magidor, M., and Shelah, S., On the strength of the interpretation method, this Journal, vol. 54 (1989), pp. 305323.
[6]Gurevich, Y. and Shelah, S., Monadic theory of order and topology in ZFC, Annals of Mathematical Logic, vol. 23 (1982), pp. 179182.
[7]Gurevich, Y. and Shelah, S., Interpreting second-order logic in the monadic theory of order, this Journal, vol. 48 (1983), pp. 816828.
[8]Gurevich, Y. and Shelah, S., The monadic theory and the ‘next world’, Israel Journal of Mathematics, vol. 49 (1984), pp. 5568.
[9]Lifsches, S. and Shelah, S., Random graphs in the monadic theory of order, in preparation.
[10]Shelah, S., The monadic theory of order, Annals of Mathematics, vol. 102 (1975), pp. 379419.
[11]Shelah, S., Notes on monadic logic, Part B: Complexity of linear orders, Israel Journal of Mathematics, vol. 69 (1990), pp. 99116.
[12]Tarski, A., Mostowski, A., and Robinson, R. M., Undecidable theories, North-Holland Publishing Company, 1953, pp. xi and 98.

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

  • Shmuel Lifsches (a1) and Saharon Shelah (a2)

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