End extensions, cofinal extensions

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Isolated gaps

For aMPA, the gap of a in M, gapM(a) is {KendM:aK}{KendM:aK}.

If McofN and bNM, then gapN(b) is non-isolated if there are d<gapN(b)<eN such that [d,e]M=, otherwise gapN(b) is isolated.


It is known that if McofN and M is recursively saturated, then the extension has non-isolated gaps.


Problem: Are there recursively saturated M and N such that McofN 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 McofN and bNM, let Mb=sup.

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?