The discussion centers on the necessity of invoking the Axiom of Choice (AC) in the context of differentiable manifolds and their associated metrics. Participants argue that while every differentiable manifold admits a Riemannian metric, the proof of this theorem does not require AC due to the properties inherent in the definition of differentiable manifolds, specifically their countable basis. The conversation also explores the implications of using a pseudoriemannian metric and the challenges of defining lengths of curves without a canonical choice of inner product, suggesting that weaker forms of choice, such as countable choice (CC), may be sufficient in certain contexts.
PREREQUISITESMathematicians, particularly those specializing in differential geometry, topology, and mathematical physics, will benefit from this discussion, as it addresses foundational concepts and the implications of choice axioms in geometric contexts.
I'm aware of this proof, it was published last year. It is quite straightforward for the vacuum equations. There is no rigorous constructive proof yet for the full EFE that I know of, though.lavinia said:
That's too bad, this site doesn't consider valid such claims without any reference.martinbn said:I don't have any references or names. I saw the discussion at the end of a talk once.
That was for Balboa, but it's ok. ;)WWGD said:Eye of the Tiger, Marciano. Eye of the Tiger!
Thanks for the reference but it is pretty useless. It is a bad sign when a ten year old preprint hasn't been published in a peer-reviewed journal. The basic argument the author use to avoid the axiom of choice is to recurr to a choice of three-dimensional geometry and giving up coordinate independence in 4 dimensions.lavinia said:
Excuse me! Are you trying to be patronizing. Arrogance is something you need to earn.RockyMarciano said:That's too bad, this site doesn't consider valid such claims without any reference.
And I guess you think you've earned it.martinbn said:Excuse me! Are you trying to be patronizing. Arrogance is something you need to earn.