Difference between revisions of "End extensions, cofinal extensions"
From Peano's Parlour
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For a∈M⊨PA, the gap of a in M, gapM(a) is ⋂{K≺endM:a∈K∧}∖⋃{K≺endM:a∉K}. | 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''. | + | If M≺cofN and b∈N∖M, then ${\rm gap}^N(b)is″non−isolated″ifthereared<{\rm gap}^N(b)<e\in Nsuchthat[d,e]\cap M=\emptyset,otherwise{\rm gap}^N(b)$ is ''isolated''. |
Revision as of 12:09, 18 January 2013
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 gaps?
Reference: Kossak, Roman; Kotlarski, Henryk More on extending automorphisms of models of Peano arithmetic. Fund. Math. 200 (2008), no. 2, 133–143.
