## 116b- Lecture 20

Given any complete, consistent extension $T$ of ${\sf PA}$, we showed that there is a minimal model $K_T$ of $T$. This model is unique up to (unique) isomorphism, it is rigid (i.e., it has no automorphisms other than the identity), and has no proper elementary substructures.

Given a model $M\models{\sf PA}$, let $SSy(M)$, the standard system of $M$, be the set of those $A\subseteq{\Bbb N}$ coded by elements of $M$, where $a\in M$ codes $A$ iff $A=\{i\in{\Bbb N}:(a)_i\ne0\}$. Thus $SSy({\Bbb N})$ is the class of finite sets. We showed that if $M\models{\sf PA}$ is nonstandard, $SSy(M)$ contains all recursive sets, and that for any non-recursive $S$ there is a nonstandard $M\models{\sf PA}$ such that $S\notin SSy(M)$.