Saturated models
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The automorphism group
In a recent paper Nurkhaidarov and Schmerl 2011 proved the following: Let κ be regular, uncounale, and such that κ<κ=κ. For each completion T⊇\PA let MκT be the saturated model of T of cardinality κ. There is a set T of completions of PA, such that |T|=2ℵ0 and for all T,T′∈T, if T≠T′, then \aut(MκT)≇\aut(MκT′).