On Guaspari's problem about partially conservative sentences

Taishi Kurahashi; Yuya Okawa; V. Y. Shavrukov; A Visser

doi:10.1016/j.apal.2022.103087

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 journal › Article › Academic › peer-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 {T_i}_i≤k of consistent r.e. extensions of Peano Arithmetic, we give a necessary and sufficient condition for the existence of a Π_n sentence which is unprovable in T_i and Σ_n-conservative over T_i for all i≤k. Secondly, we prove that for any finite family of such theories, there exists a Σ_n sentence which is simultaneously unprovable and Π_n-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 language	English
Article number	103087
Pages (from-to)	1-25
Journal	Annals of Pure and Applied Logic
Volume	173
Issue number	5
DOIs	https://doi.org/10.1016/j.apal.2022.103087
Publication status	Published - May 2022

Bibliographical note

Funding Information:
This work was supported by JSPS KAKENHI Grant Number JP19K14586.

Publisher Copyright:
© 2022 Elsevier B.V.

Keywords

Incompleteness theorem
Partial conservativity

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10.1016/j.apal.2022.103087

1-s2.0-S016800722200001X-mainFinal published version, 524 KBLicence: Taverne

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title = "On Guaspari's problem about partially conservative sentences",

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.",

keywords = "Incompleteness theorem, Partial conservativity",

author = "Taishi Kurahashi and Yuya Okawa and Shavrukov, \{V. Y.\} and A Visser",

note = "Funding Information: This work was supported by JSPS KAKENHI Grant Number JP19K14586. Publisher Copyright: {\textcopyright} 2022 Elsevier B.V.",

year = "2022",

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doi = "10.1016/j.apal.2022.103087",

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T1 - On Guaspari's problem about partially conservative sentences

AU - Kurahashi, Taishi

AU - Okawa, Yuya

AU - Shavrukov, V. Y.

AU - Visser, A

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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 - https://www.scopus.com/pages/publications/85122629579

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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 -

On Guaspari's problem about partially conservative sentences

Abstract

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