부산대학교 도서관

One of the most outstanding open questions in model theory of weaksystems of arithmetic is whether every countable model $M$ of$B\Sigma_1$ has a proper end extension to a model of $I\Delta_0$. A.~J.Wilkie and J.~B. Paris [in {\it Logic, methodology and philosophyof science, VIII (Moscow, 1987)}, 143--161, Stud. Logic Found. Math.,126, North-Holland, Amsterdam, 1989; MR1034559] showedthat the answer is positive under the additional assumption of$\Delta_0$-fullness of $M$. $\Delta_0$-fullness is a somewhat technicalcondition, implied by several other more common assumptions, such as$\Pi_1$-short recursive saturation, or totality of the exponentialfunction. Dimitracopoulos and Paschalis give a survey of the knownresults, and then, following a remark from the paper by Wilkie andParis, they give a new proof that every countable model of$B\Sigma_1+\ssf{exp}$ has a proper end extension to a model of$I\Delta_0$. Their proof is a careful application of the ArithmetizedCompleteness Theorem formulated for the notion of tableau consistency.In the last section of the paper, the authors modify their proof toshow that the assumption $\ssf{exp}$ can be replaced by any of thefollowing: (i) $M$ is $\Pi_1$-short recursively saturated; (ii) the$\Delta_0$-hierarchy provably collapses in $I\Delta_0$, and for somenonstandard $\gamma$, $M\models \forall x \exists y [y=x^\gamma]$;(iii) the $\Delta_0$-hierarchy provably collapses in $I\Delta_0$, andthere is an $a$ in $M$ such that for all $b$, $M\models b\leq a^n$, forsome standard $n$.