End extensions, cofinal extensions
Isolated gaps
For a∈M⊨PA, the gap of a in M, gapM(a) is ⋂{K≺endM:a∈K∧}∖⋃{K≺endM:a∉K}.
If M≺cofN and b∈N∖M, then gapN(b) is non-isolated if there are d<gapN(b)<e∈N such that [d,e]∩M=∅, otherwise gapN(b) is isolated.
It is known that if M≺cofN and M is recursively saturated, then the extension has non-isolated gaps.
Problem: Are there recursively saturated M and N such that M≺cofN 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≺cofN and b∈N∖M, 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?