The discussion revolves around the necessity of invoking the axiom of choice in the context of defining the length of curves on differentiable manifolds, particularly when the manifold has a standard topology that is not induced by a metric structure. Participants explore the implications of this scenario in differential geometry and mathematical physics.
Participants do not reach a consensus on whether the axiom of choice is necessary in the discussed context. Multiple competing views remain regarding the implications of the axiom of choice in differential geometry and the specific scenarios presented.
Some participants note that the axiom of choice may be hidden within certain results used in the proofs related to differentiable manifolds, and there is uncertainty about the assumptions required in the definitions of these manifolds.
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.