Difference between revisions of "Automorphisms of countable recursively saturated models"
| Line 2: | Line 2: | ||
| − | Let $G$ be the automorphism group of a countable recursively saturated model of $PA$. For every nontrivial $f\in G$ there is a countable recursively saturated elementary end extension $N$ such that $f$ has no extension to an automorphism of $N$. It is open whether there is an $f\in G$ such that for all countable recursively saturated elementary end extensions $N$ $f$ does not extend to an automorphism of $N$? | + | Let $G$ be the automorphism group of a countable recursively saturated model of $PA$. For every nontrivial $f\in G$ there is a countable recursively saturated elementary end extension $N$ such that $f$ has no extension to an automorphism of $N$. It is open whether there is an $f\in G$ such that for all countable recursively saturated elementary end extensions $N$, $f$ does not extend to an automorphism of $N$? |
If $S$ is an inductive partial satisfaction class of $M$, and $f\in{\rm Aut}(M,S)$, then there is a countable recursively saturated elementary end extension $N$ such that $f$ extends to an automorphism of $N$. If $M$ is arithmetically saturated, then there are $f\in G$ such that for every inductive partial satisfaction class $S$ of $M$, $f\notin {\rm Aut}(M,S)$. Problem: what if $M$ is not arithmetically saturated? | If $S$ is an inductive partial satisfaction class of $M$, and $f\in{\rm Aut}(M,S)$, then there is a countable recursively saturated elementary end extension $N$ such that $f$ extends to an automorphism of $N$. If $M$ is arithmetically saturated, then there are $f\in G$ such that for every inductive partial satisfaction class $S$ of $M$, $f\notin {\rm Aut}(M,S)$. Problem: what if $M$ is not arithmetically saturated? | ||
Revision as of 09:48, 18 January 2013
Extending automorphisms
Let $G$ be the automorphism group of a countable recursively saturated model of $PA$. For every nontrivial $f\in G$ there is a countable recursively saturated elementary end extension $N$ such that $f$ has no extension to an automorphism of $N$. It is open whether there is an $f\in G$ such that for all countable recursively saturated elementary end extensions $N$, $f$ does not extend to an automorphism of $N$?
If $S$ is an inductive partial satisfaction class of $M$, and $f\in{\rm Aut}(M,S)$, then there is a countable recursively saturated elementary end extension $N$ such that $f$ extends to an automorphism of $N$. If $M$ is arithmetically saturated, then there are $f\in G$ such that for every inductive partial satisfaction class $S$ of $M$, $f\notin {\rm Aut}(M,S)$. Problem: what if $M$ is not arithmetically saturated?
