Difference between revisions of "End extensions, cofinal extensions"
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Reference: Kossak, Roman; Kotlarski, Henryk ''More on extending automorphisms of models of Peano arithmetic''. Fund. Math. 200 (2008), no. 2, 133–143. | Reference: Kossak, Roman; Kotlarski, Henryk ''More on extending automorphisms of models of Peano arithmetic''. Fund. Math. 200 (2008), no. 2, 133–143. | ||
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+ | == Conservative cofinal extensions? == | ||
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+ | For M≺cofN and b∈N∖M, let Mb=sup([0,b]∩M). | ||
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+ | Cofinal extension of models of PA are never conservative, still we can define the following. A cofinal extension M≺cofN is ''conservative'' if, for each b∈N∖M there is a∈M such that | ||
+ | b∩Mb=a∩Mb. | ||
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+ | Problem: Do conservative cofinal extensions exist? |
Revision as of 12:14, 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 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([0,b]∩M).
Cofinal extension of models of PA are never conservative, still we can define the following. A cofinal extension M≺cofN is conservative if, for each b∈N∖M there is a∈M such that b∩Mb=a∩Mb.
Problem: Do conservative cofinal extensions exist?