# Can axioms of the Euclidean space be proven in the Real space?

I'm not a mathematical logic student so my question can be naïve. I'm thinking if we identify $\Bbb{E}^2$ with $\Bbb{R}^2$( as a normed space with the Euclidean norm), Then can we prove all Euclidean axioms (Hilbert's axioms) by the axioms of the real number system? If so what does it say about the completeness of the real analysis?

• Generally and very loosely my understanding is that $\mathbb{E}^n$ and $\mathbb{R}^n$ are interchangeable. One uses $\mathbb{E}^n$ to reference the space when one wants to stress its Euclidean properties. Dec 6, 2013 at 19:28
• The answer to the first question is yes (and straightforward, though partly laborious). I don't really know what your second questoin means. Dec 6, 2013 at 19:30

Once you have that, the Tarski's axioms follow from the axioms of the real field, since by Godel's completeness theorem any true statement has a proof, and the theory of real numbers is complete and is satisfied by the usual $({\bf R},+,\cdot,0,1)$, which also has all the properties postulated by Tarski under our interpretation (because, of course, ${\bf R}^3$ certainly is an euclidean space!). Alternatively, you can with some effort prove each axiom, but this can be a little tricky because of the axiom schema of continuity, which actually is an infinite set of axioms, but not all that hard.