id: 05868005
dt: j
an: 05868005
au: Lindström, Per
ti: A theorem on partial conservativity in arithmetic.
so: J. Symb. Log. 76, No. 1, 341-347 (2011).
py: 2011
pu: Association for Symbolic Logic, Poughkeepsie, NY
la: EN
cc: 
ut: elementary arithmetic; partially conservative extension
ci: Zbl 0417.03030; Zbl 1036.03002
li: doi:10.2178/jsl/1294171003
ab: The notion of partial conservativity goes back to a paper of {\it 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
    [{\it 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
    $Π_n$-sentences: one can effectively associate to each rational number
    $r$ of the interval $[0,1]$ a family $\{φ_r \mid r\in [0,1]\cap
    \mathbb{Q}\}$ of $Σ_n$-conservative $Π_n$ sentences, which increase
    in strength as $r$ decreases, satisfying the requirement that
    $\negφ_p$ is $Π_n$-conservative over $\text{PA}+φ_q$ whenever $p
    <q$. It is also possible to construct a family of $Σ_n$-sentences with
    properties as above except that the roles of $Σ_n$ and $Π_n$ 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 Indeed, the main theorem implies a yet unpublished result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.
rv: Andrea Cantini (Firenze)