Suppose $\mathbb{P}^2(k)$ is a projective plane defined/coordinatized over a commutative field $k$. Is the first-order logic of the plane completely determined by the first-order logic of $k$ ? (In other words, if $k$ and $k'$ have the same first-order logic, then what about $\mathbb{P}^2(k)$ and $\mathbb{P}^2(k')$ ?)
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2$\begingroup$ Yes, because $\mathbb P^2(k)$ has an interpretation in $k$, uniform in $k$. $\endgroup$– Emil JeřábekCommented Apr 9, 2019 at 6:35
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If by "first-order logic of the plane" you mean the theory consisting of those statements which can be phrased in terms of some ambient notion of adjacency (e.g. "point is on a line", "line A meets line B", ...)
Then the answer should be yes; since these "natural" notions of adjacency can be reduced to the existence of solutions to sets of equations built up from the operations of the field.
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2$\begingroup$ The more subtle part is that, conversely, the operations of the field can be reconstructed in terms of the incidence relation on the projective plane (see any proof of the "Fundamental theorem of projective geometry", which answers the same question up to isomorphism, not elementary equivalence). $\endgroup$ Commented Apr 9, 2019 at 19:24