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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 Indeed, the main theorem implies a yet unpublished result by Solovay and Shavrukov characterizing the isomorphism type of branches of \(E_T\) to the extent that every branch of \(E_T\) has a subset isomorphic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.

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