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Suppose we put a poset structure on the set of countable models of Peano Arithmetic as follows: for models and , let ≤ if is isomorphic to a submodel of . As described, this is just a preset (since non-isomorphic models might contain each other), but we can naturally construct a poset from it.

What can we say about the automorphism group of this poset? Is it known whether it is trivial or not?

The standard model of arithmetic is a fixed point of the action on this poset, being the unique smallest model. Are there any other fixed points? Are there interesting things we can say about this action in general? I haven't studied model theory, just read some interesting things online, so I apologize if this question is ill-posed. Further resources to study on this topic would be welcome!

Beren Gunsolus
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    No idea, but there is a book "Models of Peano Arithmetic" by Richard Kaye https://www.amazon.co.uk/Models-Peano-Arithmetic-Oxford-Guides/dp/019853213X which would seem to be a good place to look. – Angina Seng Apr 17 '20 at 05:51
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    I think you want your ordering to be "is isomorphic to a submodel of." Otherwise, you do indeed get a partial order (since $A\subseteq B$ and $B\subseteq A$ implies $A=B$). – Noah Schweber Apr 17 '20 at 14:30

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