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∑2 Induction and infinite injury priority arguments, part III: Prompt sets, minimal pairs and Shoenfield’s Conjecture

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Abstract

We prove that in everyB∑ 2 model (one satisfies ∑2 collection axioms but not ∑2 induction), every recursively enumerable (r.e.) set is either prompt or recursive. Consequently, over the base theory ∑2 collection, the existence of r.e. minimal pairs is equivalent to ∑2 induction. We also refute Shoenfield’s Conjecture inB∑ 2 models.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, National University of Singapore, Lower Kent Ridge Road, 119260, Singapore

    C. T. Chong

  2. Department of Mathematics, Faculty of Science, National University of Singapore, Lower Kent Ridge Road, 119260, Singapore

    Lei Qian

  3. Department of Mathematics, University of California Berkeley, 94720, Berkeley, CA, USA

    Theodore A. Slaman

  4. Department of Mathematics, Faculty of Science, National University of Singapore, Lower Kent Ridge Road, 119260, Singapore

    Yue Yang

Authors
  1. C. T. Chong
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  2. Lei Qian
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  3. Theodore A. Slaman
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  4. Yue Yang
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Corresponding author

Correspondence to C. T. Chong.

Additional information

This joint work was done when Theodore A. Slaman visited National University of Singapore as a visiting professor in May, 1997. Slaman was partially supported by National Science Foundation Grant DMS-9500878.

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Chong, C.T., Qian, L., Slaman, T.A. et al. ∑2 Induction and infinite injury priority arguments, part III: Prompt sets, minimal pairs and Shoenfield’s Conjecture. Isr. J. Math. 121, 1–28 (2001). https://doi.org/10.1007/BF02802493

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  • DOI: https://doi.org/10.1007/BF02802493

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