Pretty much every text about non-euclidean geometries talks about the various models by Beltrami, Riemann, Poincaré, Klein, and others which demonstrate that if euclidean geometry is consistent, then hyperbolic and elliptic geometry also are consistent. So far, so good. I think I understand these arguments.
However, there are many sources which claim that it was also proved (by Klein?) that hyperbolic geometry and euclidean geometry are equiconsistent. My understanding is that this means that if hyperbolic geometry is consistent, so is euclidean - which would place both on an equal footing.
My questions now are:
- Is this also true for elliptic geometry? Does the consistency of euclidean geometry follow from consistency of elliptic geometry?
- Where can accessible proofs for these equiconsistency results be found? (I mean proofs that not only show the well-known direction described in my first paragraph.)