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Δ2 degrees without Σ1 induction

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Abstract

In this paper, we study the structure of Turing degrees below 0′ in the theory that is a fragment of Peano arithmetic without Σ1 induction, with special focus on proper d-r.e. degrees and non-r.e. degrees. We prove:

  1. (1)

    P − + BΣ1 + Exp ⊦ There is a proper d-r.e. degree.

  2. (2)

    P − +BΣ1+ Exp ⊦ IΣ1 ↔ There is a proper d-r.e. degree below 0′.

  3. (3)

    P − + BΣ1 + Exp ⊬ There is a non-r.e. degree below 0′.

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References

  1. C. T. Chong and K. J. Mourad, The degree of a Σn cut, Annals of Pure and Applied Logic 48 (1990), 227–235.

    Article  MathSciNet  MATH  Google Scholar 

  2. C. T. Chong and K. J. Mourad, Σn definable sets without Σn induction, Transactions of the American Mathematical Society 334 (1992), 349–363.

    MathSciNet  MATH  Google Scholar 

  3. C. T. Chong and Y. Yang, Recursion theory on weak fragments of Peano arithmetic: a study of definable cuts, in Proceedings of the Aixth Asian Logic Conference (Beijing, 1996), Worl Scientific Publishing, River Edge, NJ, 1998, pp. 47–65.

    Chapter  Google Scholar 

  4. C. T. Chong and Y. Yang, The jump of a Σn cut, Journal of the London Mathematical Society 75 (2007), 690–704.

    Article  MathSciNet  MATH  Google Scholar 

  5. S. B. Cooper, Degrees of Unsolvability, Ph.D. Thesis, Leicester University, 1971.

  6. M. J. Groszek and T. A. Slaman, On Turing reducibility, preprint, 1994.

  7. R. Kaye, Models of Peano Arithmetic, Oxford University Press, 1991.

  8. R. Kaye, Model-theoretic properties characterizing Peano arithmetic, Journal of Symbolic Logic 56 (1991), 949–963.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Mytilinaios, Finite injury and Σ1-induction, Journal of Symbolic Logic 54 (1989), 38–49.

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Mytilinaios and T. Slaman, Σ2-collection and the infinite injury priority method, Journal of Symbolic Logic 53 (1988), 212–221.

    MathSciNet  MATH  Google Scholar 

  11. J. B. Paris and L. A. S. Kirby, Σn-collection schemas in arithmetic, in Logic Colloquium’ 77, Studies in Logic and the Foundations of Mathematics, Vol. 96, North-Holland, Amsterdam, 1978, pp. 199–209.

    Chapter  Google Scholar 

  12. C. Parsons, On a number theoretic choice schema and its relation to induction, in Intuitonism and Proof Theory (Proc. Conf. Bufallo. NY, 1968), North-Holland, Amstaerdam, 1970, PP. 459–474.

    Chapter  Google Scholar 

  13. T. A. Slaman, Σn-bounding and Δn-induction, Proceedings of the American Mathematical Society 132 (2004), 2449–2456.

    Article  MathSciNet  MATH  Google Scholar 

  14. T. A. Slaman and W. H. Woodin, Σ1 collection and the finite injury priority method, in Mathematical Logic and its Applications (Kyoto, 1987), Lecture Notes in Mathematics, Vol. 1388, Springer, Berlin, 1989, pp. 178–188.

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, National University of Singapore, Singapore, 119076, Singapore

    Wei Li

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  1. Wei Li
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Correspondence to Wei Li.

Additional information

The contents of this paper form a part of the author’s Ph.D. thesis at the National University of Singapore. The author is partially supported by NUS research grant WBS 146-000-159-112.

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Li, W. Δ2 degrees without Σ1 induction. Isr. J. Math. 201, 989–1012 (2014). https://doi.org/10.1007/s11856-014-1039-x

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  • DOI: https://doi.org/10.1007/s11856-014-1039-x

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