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  McofN and bNM, 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 McofN is ''conservative'' if, for each bNM there is aM such that
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bMb=aMb.
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Problem: Do conservative cofinal extensions exist?

Revision as of 12:14, 18 January 2013

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([0,b]M).

Cofinal extension of models of PA are never conservative, still we can define the following. A cofinal extension McofN is conservative if, for each bNM there is aM such that bMb=aMb.

Problem: Do conservative cofinal extensions exist?