Abstract

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 M₁, M₂ are countable arithmetically saturated models of Peano Arithmetic such that Aut(M₁) ≅ Aut( M₂), then (ω, Rep(Th(M₁))) [special characters omitted] iff (ω, Rep(Th(M₂)) [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 M₀, M₁, M₂, M₃ 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].