Difference between revisions of "Automorphisms groups of countable recursively saturated models"
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== How many automorphism groups are there? == | == How many automorphism groups are there? == | ||
| − | The main result of Kossak, Roman; Schmerl, James H. The automorphism group of an arithmetically saturated model of Peano arithmetic.J. London Math. Soc. (2) 52 (1995), no. 2, 235–244, is that if $M$ and $N$ are countable arithmetically saturated models of the same completion of $PA$, then $M\cong N$ iff ${\rm Aut}(M)\cong {\rm Aut(N)$. It is open whether the same is true for recursively saturated models which are not arithmetically saturated. It is even open whether there are nonisomorphic countable recursively saturated models whose automorphism groups are isomorphic. | + | The main result of Kossak, Roman; Schmerl, James H. The automorphism group of an arithmetically saturated model of Peano arithmetic.J. London Math. Soc. (2) 52 (1995), no. 2, 235–244, is that if $M$ and $N$ are countable arithmetically saturated models of the same completion of $PA$, then $M\cong N$ iff ${\rm Aut}(M)\cong {\rm Aut}(N)$. It is open whether the same is true for recursively saturated models which are not arithmetically saturated. It is even open whether there are nonisomorphic countable recursively saturated models whose automorphism groups are isomorphic. |
Revision as of 17:00, 17 January 2013
Generic automorphisms and the small index property
Lascar showed that countable arithmetically saturated models of $PA$ have generic automorphisms and used them to show that automorphism groups of these models have the small index property. Lascar, Daniel, The small index property and recursively saturated models of Peano arithmetic. Automorphisms of first-order structures, 281–292, Oxford Sci. Publ., Oxford Univ. Press, New York, 1994.
Do automorphism groups of countable recursively saturated, but nor arithmetically saturated models of $PA$ have generic automorphisms? Do they have the small index property.
It is shown in Kossak, Roman; Kotlarski, Henryk; Schmerl, James H. On maximal subgroups of the automorphism group of a countable recursively saturated model of PA. Ann. Pure Appl. Logic 65 (1993), no. 2, 125–148, that every open subgroup of the automorphism group of a countable recursively saturated model of $PA$ extends to a maximal subgroup. Is the same true for countable models which are recursively saturated, but not arithmetically saturated?
How many automorphism groups are there?
The main result of Kossak, Roman; Schmerl, James H. The automorphism group of an arithmetically saturated model of Peano arithmetic.J. London Math. Soc. (2) 52 (1995), no. 2, 235–244, is that if $M$ and $N$ are countable arithmetically saturated models of the same completion of $PA$, then $M\cong N$ iff ${\rm Aut}(M)\cong {\rm Aut}(N)$. It is open whether the same is true for recursively saturated models which are not arithmetically saturated. It is even open whether there are nonisomorphic countable recursively saturated models whose automorphism groups are isomorphic.
