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}\$.