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


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 $I\Sigma_{n+1}$.