On Guaspari's problem about partially conservative sentences

Taishi Kurahashi, Yuya Okawa*, V. Y. Shavrukov, A Visser

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

We investigate sentences which are simultaneously partially conservative over several theories. First, we generalize Bennet's results on this topic to the case of more than two theories. In particular, for any finite family {Ti}i≤k of consistent r.e. extensions of Peano Arithmetic, we give a necessary and sufficient condition for the existence of a Πn sentence which is unprovable in Ti and Σn-conservative over Ti for all i≤k. Secondly, we prove that for any finite family of such theories, there exists a Σn sentence which is simultaneously unprovable and Πn-conservative over each of these theories. This constitutes a positive solution to a particular case of Guaspari's problem. Finally, we demonstrate several non-implications among related properties of families of theories.

Original languageEnglish
Article number103087
Pages (from-to)1-25
JournalAnnals of Pure and Applied Logic
Volume173
Issue number5
DOIs
Publication statusPublished - May 2022

Keywords

  • Incompleteness theorem
  • Partial conservativity

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