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 Indeed, the main theorem implies a yet unpublished result by Solovay and Shavrukov characterizing the isomorphism type of branches of $$E_T$$ to the extent that every branch of $$E_T$$ has a subset isomorphic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

### Citations:

Zbl 0417.03030; Zbl 1036.03002
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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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