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The Sacks density theorem and Σ2-bounding

  • Marcia J. Groszek (a1), Michael E. Mytilinaios (a2) and Theodore A. Slaman (a3)

Abstract

The Sacks Density Theorem [7] states that the Turing degrees of the recursively enumerable sets are dense. We show that the Density Theorem holds in every model of P + BΣ2. The proof has two components: a lemma that in any model of P + BΣ2, if B is recursively enumerable and incomplete then IΣ1 holds relative to B and an adaptation of Shore's [9] blocking technique in α-recursion theory to models of arithmetic.

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[1]Chong, C. T. and Mourad, K. J., Σn definability without Σn induction, Transactions of the American Mathematical Society, vol. 334 (1992), pp. 349363.
[2]Groszek, M. J. and Mytilinaios, M. E., Σ2-induction and the construction of a high degree, Recursion theory week, Oberwolfach 1989 (Ambos-Spies, K., Muller, G. H., and Sacks, G. E., editors), Lecture Notes in Mathematics, vol. 1432, Springer-Verlag, New York, 1990, pp. 205223.
[3]Groszek, M. J. and Slaman, T. A., On Turing reducibility, preprint, 1994.
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[5]Mytilinaios, M. E. and Slaman, T. A., Σ2-collection and the infinite injury priority method, this Journal, vol. 53 (1988), pp. 212221.
[6]Sacks, G. E., On the degrees less than 0′, Annals of Mathematics, vol. 77 (1963), pp. 211231.
[7]Sacks, G. E., The recursively enumerable degrees are dense, Annals of Mathematics, vol. 80 (1964), pp. 300312.
[8]Sacks, G. E. and Simpson, S. G., The α-finite injury method, Annals of Mathematical Logic, vol. 4 (1972), pp. 343367.
[9]Shore, R. A., The recursively enumerable α-degrees are dense, Annals of Mathematical Logic, vol. 9 (1976), pp. 123155.
[10]Slaman, T. A. and Woodin, W. H., Σ-collection and the finite injury method, Mathematical logic and applications (Shinoda, J., Slaman, T. A., and Tugué, T., editors), Springer-Verlag, Heidelberg, 1989, pp. 178188.
[11]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, Heidelberg, 1987.

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