Automorphism Groups of Countable Arithmetically Saturated Models of Peano Arithmetic

Abstract

Abstract. If M, N are countable, arithmetically saturated models of Peano Arithmetic and Aut(M) ∼ = Aut(N ), then the Turingjumps of Th(M) and Th(N ) are recursively equivalent.Since 1991, when the question Are there countable, recursively saturated models M, N of PA such that Aut(M) ∼ = Aut(N ) (as abstract groups)? appeared in [8], it has been of interest to determine to what extent (the isomorphism type of) the group Aut(M) of all automorphisms of a countable, recursively saturated model M of Peano Arithmetic determines (the isomorphism type of) M. It was proved in [8] that whenever both M and N are countable, recursively saturated models of PA and exactly one of them is arithmetically saturated, then Aut(M) and Aut(N ) are not isomorphic as topological groups. In 1994, Lascar [16] proved that countable, arithmetically saturated models of PA have the small index property, and that result then implied that Aut(M) ∼ = Aut(N ) as abstract groups. This gave the first positive answer to the above question. A neater way, in which the use of the small index property is masked, that automorphism groups distinguish those models that are arithmetically saturated from the other countable recursively saturated ones was obtained the next year in [13, Coro. 3.9] (or see [15, Th. 9.3.10]): If M is countable and recursively saturated, then M is arithmetically saturated iff the cofinality of Aut(M) is uncountable. Finally, we mention that Kaye's Theorem [7] (see §1) characterizing the closed normal subgroups of Aut(M), which appeared in the same volume [9] as did Lascar's Theorem, yields that whenever M, N are countable, arithmetically saturated models and M is a model of True Arithmetic (TA) while N is not, then Aut(M) ∼ = Aut(N ).

Decoding in the automorphism group of a recursively saturated model of arithmetic

“…In a recent paper the following result is proven. Theorem There is a set of countable arithmetically saturated models of such that whenever , then and .…”