Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T11:39:10.025Z Has data issue: false hasContentIssue false
  • English
  • Français

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]
Get access Rights & Permissions [Opens in a new window]

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 The latter result allows to re-obtain an unpublished result of Solovay, the presence of a subset order-isomorphic to the reals in every non-trivial end-segment of every branch of the E-tree, and to generalize it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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Arana, A., Arithmetical independence results using higher recursion theory, this Journal, vol. 69 (2004), pp. 1–8.Google Scholar
[2]Guaspari, D., Partially conservative extensions of arithmetic, Transactions of the American Mathematical Society, vol. 254 (1979), pp. 47–68.CrossRefGoogle Scholar
[3]Jensen, D. and Ehrenfeucht, A., Some problem in elementary arithmetics, Fundamenta Mathematicae, vol. 92 (1976), pp. 223–245.CrossRefGoogle Scholar
[4]Lindström, P., Aspects of incompleteness, 2nd ed., Lecture Notes in Logic, vol. 10, A K Peters, Natick, 2003.Google Scholar
[5]Misercque, D., Sur le treillis des formuées fermies untverselles de l'arithmétique de Peano, Ph.D. thesis, Université Libre de Bruxelles, 1985.Google Scholar
[6]V. Yu., Shavrukov, Dense chains of Σn sentences with strong conservativity properties, in preparation.Google Scholar
[7]Shavrukov, V. Yu. and Solovay, R. M., Branches of the E-tree and jump pseudo-hierarchies, in preparation.Google Scholar