When studying the automorphism group Aut( M ) of a model M , one is interested to what extent M is recoverable from Aut( M ). We show that if M is a countable arithmetically saturated of Peano Arithmetic, then Aut( M ) can recognize if a maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element.
We use that fact to show that if M 1 , M 2 are countable arithmetically saturated models of Peano Arithmetic such that Aut( M 1 ) [congruent with] Aut( M 2 ), then (ω, Rep(Th( M 1 ))) [Special characters omitted.] iff (ω, Rep(Th( M 2 )) [Special characters omitted.] . Here: [Special characters omitted.] is Ramsey's Theorem stating that every 2-coloring of [ω] n has an infinite homogeneous set; and if T ⊃ PA is a complete and consistent theory, then we define Rep( T ) = {ω ∩ X : X is a definable set in a prime model of T }.
Using this result we show the existence of countable arithmetically saturated models M 0 , M 1 , M 2 , M 3 of Peano Arithmetic such that they have the same standard system and Aut( M i ) [Special characters omitted.] Aut( M j ), where i < j < 4.
We also show the similar results for saturated models of Peano Arithmetic of cardinality [Special characters omitted.] .
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