Series: Penn State Logic Seminar Date: Tuesday, November 5, 2002 Time: 2:30 - 3:45 PM Place: 312 Boucke Building Speaker: Roman Kossak, Mathematics, CUNY Title: Nonstandard Standard Systems Abstract: Let $M$ be a model of PA and let $I$ be an initial segment of it. The standard system of $M$ relative to $I$, denoted by $SSy(M/I)$ is the family of sets of the form $X\cap I$, where $X$ is definable in $M$ (with parameters). The most general question I want to address is: Given a model $M$ of PA and a family $S$ of its subsets, is there an elementary extension end $N$ of $M$ such that $SSy(N/M)=S$? Vladimir Kanovei proved that every countable recursively saturated model $M$ of PA has an elementary end extension $N$ such that $(M,SSy(N/M))$ is model of $I\Sigma_n$ and not $I\Sigma_{n+2}$ (in the language with predicate symbols for all elements of $SSy(N/M)$). I will outline a more straightforward proof of a slightly sharper result replacing $I\Sigma_{n+2}$ with $I\Sigma_{n+1}$.