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Tarski's axioms (see here: http://en.wikipedia.org/wiki/Tarski%27s_axioms) are a first-order axiomatization of Euclidean Geometry. Now, I believe the standard model for the axioms is the real number plane (for 2D plane geometry) and 3D space (for the 3D form) with "congruence of segments" and "betweenness" defined in the usual ways via the distance formula. But what does a non-standard model of the axioms look like? I'm especially curious about how the "continuity axiom (schema)" constrains the choice.

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    $\begingroup$ I'm not familiar with the axioms, but my guess is that you can take $R^2$ where $R$ is any real closed field (en.wikipedia.org/wiki/Real_closed_field). $\endgroup$ Apr 2 '13 at 1:27
  • $\begingroup$ Yes, begin with a non-standard real-closed field. $\endgroup$
    – GEdgar
    Apr 2 '13 at 1:28
  • $\begingroup$ The coordinate geometry over the real algebraics probably deserves the name standard model. $\endgroup$ Apr 2 '13 at 1:31
  • $\begingroup$ And I'm pretty sure every model is isomorphic to the coordinate plane over a real closed field. $\endgroup$
    – user14972
    Apr 2 '13 at 1:42
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    $\begingroup$ @AndréNicolas: I'd say coordinate geometry over the reals deserve that name more. $\endgroup$ Apr 2 '13 at 1:59
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I'm going to answer my own question! I found this paper, which gives the answer. It's called "Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries" by MJ Greenberg, 2010.

http://www.maa.org/programs/maa-awards/writing-awards/old-and-new-results-in-the-foundations-of-elementary-plane-euclidean-and-non-euclidean-geometries

From the paper:

Geometrically, Tarski-elementary plane geometry certainly seems mysterious, but the representation theorem illuminates the analytic geometry underlying it: its models are all Cartesian planes coordinatized by real-closed fields.

(pg. 215) (pg. 18 of the paper)

So Qiaochu (in the comments to this question) was right, apparently.

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