Logic Seminar
Julia Knight
University of Notre Dame
Integer parts for real closed fields
Abstract: The real closed ordered fields are the models of the theory of the ordered field of reals. An integer part sits in $R$ in the way that the integers sit in the reals. More precisely, an integer part for an ordered field $R$ is a discrete ordered ring $I\subseteq R$ such that $1$ is the first positive element, and for each $x\in R$, there exists $i\in I$ such that $i\leq x < i+1$. Shepherdson showed that $I$ is an integer part for a real closed ordered field iff it is a model of $IOpen$---the fragment of arithmetic with induction axioms just for open formulas. It is natural to ask when a real closed ordered field $R$ has an integer part satisfying full $PA$. Paola D'Aquino, Sergei Starchenko, and I showed that if $R$ has an integer part $I$ which is a nonstandard model of $PA$, then $R$ is recursively saturated, with types determined by $I$. Hence, if $R$ is countable, then $I$ determines the isomorphism type. We also showed that if $R$ is recursively saturated, then there is an integer part $I$ satisfying $PA$ such that $R$ is the real closure of $I$.
Mourgues and Ressayre, with some help from Marker and Delon, showed that every real closed ordered field has an integer part. Currently, Karen Lange and I are considering the complexity of integer parts.
seminar begins with tea.
Tuesday October 28, 2008 at 4:00 PM in SEO 612