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I came across the wonderful fact that the theory of Euclidean geometry in 2 dimensions is complete, consistent and decidable — as shown by Tarski's axiomatization.

I know very little about this. My question: What about the dimensions higher than 2? Are there similar axiomatizations that preserve completeness, consistency and decidability?

I don't really see why the answer has to be "yes" (or "no").

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    $\begingroup$ Btw, any axiomatization of an existing model has to be consistent (or otherwise it simply is not an axiomatization). One interesting example that generalizes euclidean geometry is the inner product space theory. Which is known to be decidable. But I don't know whether it is complete. $\endgroup$
    – freakish
    Mar 14 at 22:00
  • $\begingroup$ Interesting. Can we get a positive result by dropping the Lower Dimension and Higher Dimension axioms, picking a reasonable model $\mathfrak{A}$ with carrier $\mathbb{R}^3$, and then taking $\text{Th}(\mathfrak{A})$? $\endgroup$
    – Greg Nisbet
    Mar 14 at 22:09

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