# What's a non-standard model of Tarskian Euclidean geometry?

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.

• 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). Apr 2 '13 at 1:27
• Yes, begin with a non-standard real-closed field. Apr 2 '13 at 1:28
• The coordinate geometry over the real algebraics probably deserves the name standard model. Apr 2 '13 at 1:31
• And I'm pretty sure every model is isomorphic to the coordinate plane over a real closed field.
– user14972
Apr 2 '13 at 1:42
• @AndréNicolas: I'd say coordinate geometry over the reals deserve that name more. Apr 2 '13 at 1:59