Difference between revisions of "End extensions, cofinal extensions"

Revision as of 12:14, 18 January 2013

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.

It is known that if $M\prec_{cof} N$ and $M$ is recursively saturated, then the extension has non-isolated gaps.

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

Reference: Kossak, Roman; Kotlarski, Henryk More on extending automorphisms of models of Peano arithmetic. Fund. Math. 200 (2008), no. 2, 133–143.

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. A cofinal 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?