TY - JOUR

T1 - On Guaspari's problem about partially conservative sentences

AU - Kurahashi, Taishi

AU - Okawa, Yuya

AU - Shavrukov, V. Y.

AU - Visser, A

N1 - Funding Information:
This work was supported by JSPS KAKENHI Grant Number JP19K14586.
Publisher Copyright:
© 2022 Elsevier B.V.

PY - 2022/5

Y1 - 2022/5

N2 - 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.

AB - 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.

KW - Incompleteness theorem

KW - Partial conservativity

UR - http://www.scopus.com/inward/record.url?scp=85122629579&partnerID=8YFLogxK

U2 - 10.1016/j.apal.2022.103087

DO - 10.1016/j.apal.2022.103087

M3 - Article

SN - 0168-0072

VL - 173

SP - 1

EP - 25

JO - Annals of Pure and Applied Logic

JF - Annals of Pure and Applied Logic

IS - 5

M1 - 103087

ER -