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A theorem on partial conservativity in arithmetic

Published online by Cambridge University Press:  12 March 2014

Per Lindström
Per Lindström*
Affiliation:
E-mail: [email protected]
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Abstract

Improving on a result of Arana, we construct an effective family (φr ∣ r ϵ ℚ ∩ [0,1]) of Σn-conservative Πn sentences, increasing in strength as r decreases, with the property that ¬φp is Πn-conservative over PA + φq whenever p < q. We also construct a family of Σn sentences with properties as above except that the roles of Σn and Πn are reversed. The latter result allows to re-obtain an unpublished result of Solovay, the presence of a subset order-isomorphic to the reals in every non-trivial end-segment of every branch of the E-tree, and to generalize it to analogues of the E-tree at higher levels of the arithmetical hierarchy.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 1 , March 2011 , pp. 341 - 347
Copyright
Copyright © Association for Symbolic Logic 2011

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References

REFERENCES

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