MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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
hi Hans. What's your email address? – Colin Tan Dec 11 '10 at 10:41
up vote 2 down vote accepted

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.

Well, I know a ton more about this at secondhand, you might say. Let me know if you need more info to get started.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.