Let $G$ be the automorphism group of the binary rooted tree.
The downward Löwenheim-Skolem theorem states that G has a countable elementary sub-structure.
My question is whether such sub-structure is explicitly known?
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asked Apr 14, 2018 at 11:17
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$\begingroup$ Is there an explicit description of the group or its first-order theory? $\endgroup$– Emil JeřábekCommented Apr 14, 2018 at 19:22
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$\begingroup$ @Emil: I think this group has a pretty explicit description as a pro-$p$ group with $p=2$, by mapping it onto the automorphisms groups of the $n$-neighborhoods of the root, for all integers $n \ge 1$. $\endgroup$– Lee MosherCommented Apr 14, 2018 at 20:46
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$\begingroup$ I don't know much about the first-order theory of that group, but the set of computable automorphisms seems extremely likely to be an elementary substructure. $\endgroup$– Alex MennenCommented Apr 22, 2018 at 6:48
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