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The Sacks density theorem and Σ2-bounding

Published online by Cambridge University Press:  12 March 2014

Marcia J. Groszek ,
Michael E. Mytilinaios  and
Theodore A. Slaman
Marcia J. Groszek
Affiliation:
Department of Mathematics, Dartmouth College, Hanover New Hampshire 03755, USA, E-mail: [email protected]
Michael E. Mytilinaios
Affiliation:
Department of Informatics, Athens University of Economics, Patission 76, Athens 104 34, Greece, E-mail: [email protected]
Theodore A. Slaman
Affiliation:
Department of Mathematics, The University of Chicago, Chicago, Illinois 60637, USA, E-mail: [email protected]
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 2 , June 1996 , pp. 450 - 467
Copyright
Copyright © Association for Symbolic Logic 1996

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References

REFERENCES

[1]Chong, C. T. and Mourad, K. J., Σn definability without Σn induction, Transactions of the American Mathematical Society, vol. 334 (1992), pp. 349–363.Google Scholar
[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. 205–223.Google Scholar
[3]Groszek, M. J. and Slaman, T. A., On Turing reducibility, preprint, 1994.Google Scholar
[4]Mytilinaios, M. E., Finite injury and Σ2-induction, this Journal, vol. 54 (1989), pp. 38–49.Google Scholar
[5]Mytilinaios, M. E. and Slaman, T. A., Σ2-collection and the infinite injury priority method, this Journal, vol. 53 (1988), pp. 212–221.Google Scholar
[6]Sacks, G. E., On the degrees less than 0′, Annals of Mathematics, vol. 77 (1963), pp. 211–231.CrossRefGoogle Scholar
[7]Sacks, G. E., The recursively enumerable degrees are dense, Annals of Mathematics, vol. 80 (1964), pp. 300–312.Google Scholar
[8]Sacks, G. E. and Simpson, S. G., The α-finite injury method, Annals of Mathematical Logic, vol. 4 (1972), pp. 343–367.CrossRefGoogle Scholar
[9]Shore, R. A., The recursively enumerable α-degrees are dense, Annals of Mathematical Logic, vol. 9 (1976), pp. 123–155.Google Scholar
[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. 178–188.Google Scholar
[11]Soare, R. I., Recursively enumerable sets and degrees, Springer-Verlag, Heidelberg, 1987.Google Scholar