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The thickness lemma from P + IΣ1 + ¬BΣ2

  • Yue Yang (a1)

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Let P− denote the Peano axioms minus the induction scheme. Let IΣn, (In), BΣn (Bn), LΣn (Ln denote the induction scheme, the collection scheme, and the least number principle for Σn-(∏n-) formulas respectively. Paris and Kirby [3] studied the relative proof-theoretic strengths of those schemes. The general theorem states that IΣn, In, LΣn, and Ln are equivalent; IΣn implies BΣn implies IΣn1; but not conversely.

In recent years, people have been interested in doing recursion theory on fragments of arithmetic. One of the purposes of this study is to understand the priority methods. Much work has been done in this area. For example, M. Mytilinaios [5] showed that the Sacks splitting theorem can be proven in P− + IΣ1. Later, J. Mourad showed that the Sacks splitting theorem is indeed equivalent to IΣ1 [4]. M. Groszek and M. Mytilinaios [1] showed that P− + IΣ2 is sufficient to prove the existence of a high incomplete r.e. set. On the other hand, M. Mytilinaios and T. Slaman [6] showed that P− + IΣ1 is too weak to prove the existence of such a set. A natural question to ask is if the existence of such a set implies IΣ2. In this paper, we will show the answer is negative by constructing a model of P− + IΣ1 + ¬BΣ2 which has a high incomplete r.e. set. Notice that, as shown by M. Groszek and T. Slaman in [2], P− + IΣ1 is too weak to show the transitivity of weak Turing reducibility on Σ2-sets.

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[1] Groszek, M. J. and Mytilinaios, M. E., Σ2-induction and the construction of a high degree, Recursion Theory Week, Proceedings Oherwolfach 1989 (Ambos-Spies, K.et al, editors), Lecture Notes in Mathematics, vol. 1432, Springer-Verlag, Berlin, 1990, pp. 205223.
[2] Groszek, M. J. and Slaman, T. A., On Turing reducibility, preprint.
[3] Kirby, L. A. S. and Paris, J. B., Σn-collection schemas in arithmetic, Logic colloquium ’77, North-Holland, Amsterdam, 1978, pp. 199209.
[4] Mourad, J., Ph.D. thesis. University of Chicago, Chicago, Illinois, 1989.
[5] Mytilinaios, M. E., Finite injury and Σ1-induction, this Journal, vol. 53 (1988), pp. 212221.
[6] Mytilinaios, M. E. and Slaman, T. A., Σ2-collection and the infinite injury priority method, this Journal, vol. 54 (1989), pp. 3849.
[7] Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1987.

The thickness lemma from P + IΣ1 + ¬BΣ2

  • Yue Yang (a1)

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