##
**A theorem on partial conservativity in arithmetic.**
*(English)*
Zbl 1218.03033

The notion of partial conservativity goes back to a paper of D. Guaspari that appeared in 1979 [“Partially conservative extensions of arithmetic”, Trans. Am. Math. Soc. 254, 47–68 (1979; Zbl 0417.03030)]. The author applies ideas from his book on incompleteness [P. Lindström, Aspects of incompleteness. 2nd ed. Lecture Notes in Logic 10. Natick, MA: Association for Symbolic Logic (2003; Zbl 1036.03002)] in order to prove a theorem on partial conservative primitive recursive consistent extensions of Peano arithmetic.

It is shown how to produce dense chains of strong conservative \(\Pi_n\)-sentences: one can effectively associate to each rational number \(r\) of the interval \([0,1]\) a family \(\{\varphi_r \mid r\in [0,1]\cap \mathbb{Q}\}\) of \(\Sigma_n\)-conservative \(\Pi_n\) sentences, which increase in strength as \(r\) decreases, satisfying the requirement that \(\neg\varphi_p\) is \(\Pi_n\)-conservative over \(\text{PA}+\varphi_q\) whenever \(p <q\). It is also possible to construct a family of \(\Sigma_n\)-sentences with properties as above except that the roles of \(\Sigma_n\) and \(\Pi_n\) are reversed.

These results have an application to the so-called \(E_T\)-tree (concerning the the lattice of universal sentences in Peano arithmetic). Indeed, the main theorem implies a yet unpublished result by Solovay and Shavrukov characterizing the isomorphism type of branches of \(E_T\) to the extent that every branch of \(E_T\) has a subset isomorphic to the real numbers.

It is shown how to produce dense chains of strong conservative \(\Pi_n\)-sentences: one can effectively associate to each rational number \(r\) of the interval \([0,1]\) a family \(\{\varphi_r \mid r\in [0,1]\cap \mathbb{Q}\}\) of \(\Sigma_n\)-conservative \(\Pi_n\) sentences, which increase in strength as \(r\) decreases, satisfying the requirement that \(\neg\varphi_p\) is \(\Pi_n\)-conservative over \(\text{PA}+\varphi_q\) whenever \(p <q\). It is also possible to construct a family of \(\Sigma_n\)-sentences with properties as above except that the roles of \(\Sigma_n\) and \(\Pi_n\) are reversed.

These results have an application to the so-called \(E_T\)-tree (concerning the the lattice of universal sentences in Peano arithmetic). Indeed, the main theorem implies a yet unpublished result by Solovay and Shavrukov characterizing the isomorphism type of branches of \(E_T\) to the extent that every branch of \(E_T\) has a subset isomorphic to the real numbers.

Reviewer: Andrea Cantini (Firenze)

### MSC:

03F30 | First-order arithmetic and fragments |

03D35 | Undecidability and degrees of sets of sentences |

03F25 | Relative consistency and interpretations |

03F40 | Gödel numberings and issues of incompleteness |

Full Text:
DOI

### References:

[1] | Aspects of incompleteness 10 (2003) |

[2] | Arithmetical independence results using higher recursion theory 69 pp 1– (2004) · Zbl 1067.03074 |

[3] | DOI: 10.1090/S0002-9947-1979-0539907-7 · doi:10.1090/S0002-9947-1979-0539907-7 |

[4] | Fundamenta Mathematicae 92 pp 223– (1976) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.