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Representation theorems for recursively enumerable sets and a conjecture related to poonen’s large subring of \(\bf \mathbb{Q} \)

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It is remarked that unsolvability results can often be extended to yield novel “representation” theorems for the set of all recursively enumerable sets. In particular, it is shown that an analysis of the proof of the unsolvability of Hilbert’s 10th problem over Poonen’s large subring of \( \mathbb{Q} \) can provide such a theorem. Moreover, applying that theorem to the case of a simple set leads to a conjecture whose truth would imply the unsolvability of Hilbert’s 10th problem over \( \mathbb{Q} \).

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References

  1. W. Boone,“The word problem,” Ann. Math., 70, 207–265 (1959).

    Article  MathSciNet  Google Scholar 

  2. G. Higman, “Subgroups of finitely presented groups,” Pro. Roy. Soc. London Ser. A, 262, 455–475 (1961).

    Article  MATH  MathSciNet  Google Scholar 

  3. Yu. V. Matiyasevich, Hilbert’s Tenth Problem, MIT Press, Cambridge, Massachusetts (1993).

    Google Scholar 

  4. S. P. Novikov, “On the algorithmic insolvability of the word problem in group theory,” Amer. Math. Soc. Transl., Ser. 2, 91, 1–122 (1958).

    Google Scholar 

  5. B. Poonen, “Hilbert’s tenth problem and Mazur’s conjeture for large subrings of \( \mathbb{Q} \),” J. Amer. Math. Soc., 16, No. 4, 981–990 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  6. E. L. Post, “Recursively enumerable sets of positive integers and their decision problems,” Bull. Amer. Math. Soc., 50, 284–316 (1944). Reprinted in: M. Davis (ed.), The Undeidable, Raven Press (1965) and Dover (2004), pp. 305–337; M. Davis (ed.), Solvability, Provability, Definability: The Collected Works of Emil L. Post, Birkhäuser (1994), pp. 461–493.

  7. J. Rotman, An Introduction to the Theory of Groups, Springer-Verlag, Berlin-New York (1994).

    Google Scholar 

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Authors and Affiliations

  1. Courant Institute, New York University, New York, USA

    M. Davis

  2. Visiting Scholar at the University of California at Berkeley, Berkeley, USA

    M. Davis

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  1. M. Davis
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Correspondence to M. Davis.

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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 377, 2010, pp. 50–54.

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Davis, M. Representation theorems for recursively enumerable sets and a conjecture related to poonen’s large subring of \(\bf \mathbb{Q} \) . J Math Sci 171, 728–730 (2010). https://doi.org/10.1007/s10958-010-0176-7

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  • DOI: https://doi.org/10.1007/s10958-010-0176-7

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