Automorphisms of countable recursively saturated models

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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 $N$ [1]?


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 recursively, but not arithmetically, saturated [1]?


Bumps

Let $M$ be a countable recursively saturated model of $PA$ and let $I$ be a strong elementary cut of $M$. An automorphism $f$ of $M$ is an $I$-bump if for all $x\in I$, $f(x)=x$ and for all $x>I$, $f(x)>x$. Bumps do exist [2].


Problem: Under the assumptions above, let $f$ and $g$ be $I$-bumps. Are $f$ and $g$ conjugate, i.e. is there an automorphism $h$ such that $f=h^{-1}gh$?


References

  1. Roman Kossak. Four problems concerning recursively saturated models of arithmetic. Notre Dame J. Formal Logic 36(4):519--530, 1995. (Special Issue: Models of arithmetic) www   DOI   MR   bibtex
  2. R. Kossak. Exercises in ‘back-and-forth’. Proceedings of the Nineth Easter Conference on Model Theory, Gosen, 1991. bibtex
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