On Sat, Aug 31, 2024 at 10:48 PM Alan Grayson <[email protected]> wrote:
* > please explain how the metric tensor can be defined unambiguously at > some point P on the underlying manifold, spacetime, if there is an > uncountable set of pairs on a vector space on the tangent space at some > point P on which the metric tensor is defined* If, as I suspect, your interest is physics and not pure mathematics then it's a non-issue. The fact is nobody is even sure that 4D space-time contains an infinite number of points, for all we know it may only contain an astronomical number to an astronomical power number of points. That's undoubtedly a very big number but it's no closer to being infinite than the number one is. And even if 4D space-time does contain an uncountabley infinite number of points, if you simplify your physical theory by assuming there is only a countably infinite number of points it will have a negligible effect on your theory; that is to say you could make the discrepancy between what your theory predicts will happen and what you actually observed to happen in experiments to be arbitrarily small. I am not aware of any physical theory in which the difference between countable infinity and uncountable infinity leads to different experimentally testable predictions. John K Clark See what's on my new list at Extropolis <https://groups.google.com/g/extropolis> n4x -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/CAJPayv2R0Zp%3DAhZZ93_nk%3DcCYab8_T1sh9s-LuZpX67c0boRag%40mail.gmail.com.
