# Reference: Countable Models of (Non-)Euclidean Geometry

Has there been a survey written on the model theory of first-order (non-)Euclidean geometry in the spirit of Hilbert and Tarski? I'm especially interested in two aspects of the model theory:

1. Countable models
2. Distinguished/canonical models.

I'm interested in 2 because I don't see why the standard ${\mathbb{R}}^2$ model is any special.

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Of course you can take the algebraic points in $\mathbb{R}^2$ to get a countable model. –  Dylan Thurston May 2 '10 at 19:05
I sent you Greenberg's article an hour or two ago. I think you will like it. –  Will Jagy May 3 '10 at 19:37
@Will/Colin: Can you send me the article, too, please? –  Hans Stricker Dec 10 '10 at 16:45

Marvin Jay Greenberg got very interested in the foundations for the fourth edition(2007) of his book. This led to a survey article in the March 2010 M.A.A. Monthly. Table of contents:

It does not seem to say explicitly on the link, volume 117, number 3, March 2010, pages 198-219

I have a pdf of the article if you have no better way to look at it, just email me. Marvin sent me a copy because my results are in it, however I was not doing genuine foundations.

The other book is by the well known R. Hartshorne, called Geometry:Euclid and Beyond (2000)

That's a pretty good start, article and two books. You may also want to look up Victor Pambuccian on MathSciNet

To summarize the bits I expect you find most interesting, Hilbert gave a recipe for defining the hyperbolic plane first and finding the underlying field second, this is called the "field of ends" in English. I prefer the "Euclidean" fields, if there is a field element $a > 0$ then $\sqrt a$ is also in the field. The smallest field that does everything I find interesting is the "constructible field" which is all numbers arrived at by starting with the rationals and taking a finite number of square roots, mixed with other field operation of course. This is a subfield of the algebraic numbers. Hilbert himself concentrated on the milder Pythagorean fields, if $a \in F$ then $\sqrt{1 + a^2} \in F.$ So there may be positive field elements without square roots. As there is an intermediate step that usually requires interpreting the standard hyperbolic functions, Hartshorne wrote that entire section with a "multiplicative length" for segments, which corresponds to taking $e^x$ when $x$ is an ordinary length in a plane over the reals. Personally, I prefer $\sinh x$ because of the appearance of the (hyperbolic) Pythagorean Theorem and the integral triangles you get this way, but that is never going to be popular.