While perusing p. 237 of the 3rd ed. of Marvin Greenberg's book on Euclidean and non-Euclidean geometries, I learned that it can actually be proven that "all possible models of hyperbolic geometry are isomorphic to one another, i.e., that the axioms for hyperbolic geometry are categorical".

This assertion made me wonder if the analogous result for the axioms for elliptic geometry holds true. Unfortunately, I couldn't find a remark regarding this issue in Greenberg's book. I believe that an answer can be found in the following reference:

W. Schwabhäuser, On models of elementary elliptic geometry. Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), pp. 312–328. North-Holland Publishing Co., Amsterdam, 1965.

Since this article is behind a paywall for me, I would like to know if anybody here has already read it and can tell me whether or not Schwabhäuser establishes the categoricity of some version of elliptic geometry therein... In case Schwabhäuser proves nothing of this sort in the said article, what would be the paper to look at in order to know the state of the art regarding the axioms for elliptic geometry?

Thanks in advance for taking the time to take a look at this question of mine.


1 Answer 1


In the mentioned article, Schwabhäuser proves that all models of elementary elliptic geometry are isomorphic to elliptic Klein spaces over real closed fields. Actually, the paper only deals with the case of elementary plane elliptic geometry and remarks that the results extend to higher dimensions with no additional difficulty. The details are a bit involved, but the gist is to show the result first for Pythagorean-ordered fields, then for Euclidean-ordered fields, and lastly for real closed fields.


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