Please help me understand this concept of completeness of geometry and set me on the right path.
This is my context: From wikipedia, a formal system is complete if every tautology is also a theorem. For me this means that a system of axioms is complete if every statement that is true can be proved from the axioms (the question is tautologies in what kind of logic, first order, second order,…?). Anyway, it seems then that the concept of completeness depend on the system of axioms.
So, in the case of Euclidean Geometry, its completeness depends on their axioms (For example Euclid's Axioms, Hilbert Axioms, Tarki's Axioms,etc). Comparing for example the axioms of Hilbert and the axioms given by Tarski, I can see that they are essentially different in that Hilbert uses second order logic and Tarski's only first order logic. Also Tarski proved that his system of axioms is complete.
Now, the incompleteness theorem of Gödel stated that the system of axioms of arithmetic is incomplete. So, when we consider $R^2$ and $R^3$ as a representation of Euclidean Geometry we are talking about a system that is not complete because we are using the real numbers. So, are we speaking of different Euclidean Geometries? I'm totally confused.
Best Answer
It sounds like you might get a lot out of Greenberg's short paper Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries.
In it he discusses how the ordinary theory of plane geometry is incomplete (pg 202), but why Tarski's geometry is complete (section 3.3, page 215).