End extensions, cofinal extensions

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Conservative cofinal extensions?

For $M\prec_{cof} N$ and $b\in N\setminus M$, let $M_b=\sup([0,b]\cap M)$.

Cofinal extension of models of $PA$ are never conservative, still we can define the following. An extension $M\prec_{cof} N$ is conservative if, for each $b\in N\setminus M$ there is $a\in M$ such that $b\cap M_b=a\cap M_b.$

Problem: Do conservative cofinal extensions exist?


The description property

The extension $M\prec_{cof} N$ has the description property if for every $a\in N\setminus M$ there is a coded in $N$ nested sequence $(A_i:i<\omega)$ of $M$-finite sets such that

1. $N\models a\in A_i$ for all $i<\omega$;

2. For each $M$-finite $B$ such that $a\in B$, there is an $i<\omega$ such that $A_i\subseteq B$.


For each countable recursively saturated $M\models PA$, there are $K$ and, $N$ such that $K\prec_{cof} M\prec_{cof} N$ and both extensions have the description property.Moreover, we can require that $K$ is recursively saturated [1].


Problem: Let $M\models PA$ be countable and recursively saturated. Is there an $N$ such that the extension $M\prec_{cof} N$ has the description property and ${\rm SSy}(M)={\rm SSy}(N)$?

Isolated gaps

For $a\in M\models PA$, the gap of $a$ in $M$, ${\rm gap}^M(a)$ is $\bigcap\{K\prec_{end} M: a\in K\land \}\setminus \bigcup\{K\prec_{end} M: a\notin K\}$.

If $M\prec_{cof} N$ and $b\in N\setminus M$, then ${\rm gap}^N(b)$ is non-isolated if there are $d<{\rm gap}^N(b)<e\in N$ such that $[d,e]\cap M=\emptyset$, otherwise ${\rm gap}^N(b)$ is isolated.

If $M\prec_{cof} N$ and $M$ is recursively saturated, then the extension has non-isolated gaps. Moreover, if the extension has the description property, then all new gaps are non-isolated [1].


Problem: Are there recursively saturated $M$ and $N$ such that $M\prec_{cof} N$ and the extension has an isolated gap?


Extending automorphisms to cofinal extensions

If $M$ and $N$ are countable recursively saturated models of $PA$, $M\prec_{cof} N$ and $f$ is an automorphism of $M$, then $f$ is potentially extendable if for every bounded $A\subseteq M$, $A$ is coded in $N$ iff $f(A)$ is.

If $M$ and $N$ are countable recursively saturated models of $PA$ and the extension $M\prec_{cof} N$ has the description property, then every potentially extendable automorphism of $M$ has an extension to an automorphism of $N$ [1].


Problem: Find a sufficient and necessary condition on the extension $M\prec_{cof} N$, where $M$ and $N$ are countable recursively saturated models of $PA$, under which every potentially extendable automorphism of $M$ extends to an automorphism of $N$.


Strong cofinal extensions?

Find a notion of strongness for cofinal extensions and prove that if If $M$ and $N$ are countable recursively saturated models of $PA$ and the extension $M\prec_{cof} N$ is strong than there is an automorphism $f$ of $N$ whose fixed point set is $M$ [1].


References

  1. Roman Kossak and Henryk Kotlarski. More on extending automorphisms of models of Peano arithmetic. Fund. Math. 200(2):133--143, 2008. www   DOI   MR   bibtex
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