On Friday, November 28, 2025 at 3:26:03 PM UTC-7 Russell Standish wrote:
On Thu, Nov 27, 2025 at 07:29:34AM -0800, Alan Grayson wrote: > > > IMO there's no way to do this, which is why we have the Axiom of Choice, but > in > this case a situation where instead of having an uncountable collection of > uncountable states, we have only a single uncountable state. So, using this > reduced situation, all we can say is that we "can" select one point on this > set, say to define an origin of coordinates, but we can't say how to select it. > AG > Sorry - I can't make sense of your question. *The Axiom of Choice (AoC) asserts that given an uncountable set of sets, each one being* *uncountable, there is a set composed of one element of each set of the uncountable set* *of sets. The AoC doesn't tell us how such a set is constructed, only that we can assume it* *exists. So, in chosing an origin for the coordinate system for a plane say, we have to apply* *the AoC for a single uncountable set, the plane. But there's no way to construct it. Does* *this make sense? AG * > > On other thing; when evaluating the tensor T(u), how do you know which > co-vector (member of dual vector space) to use, or doesn't it matter? > Won't different co-vectors result in different real values for the tensor? > AG > > > This issue remains and seems important. If we choose the transpose of u to > evaluate the tensor, we get the inner product. But is this what Einstein means > for the tensors in his field equations? AG > Yes, although typically in Relativity, one lives in "Minkowski space", ie the "inner product" is gⁱʲxᵢxⱼ where for flat spacetime, g⁰⁰=-1 and gⁱʲ=δᵢⱼ for i>0, ie the axiom of positive definiteness is discarded. The stuff about dual spaces, linear maps etc go over just fine to this slightly more general situation, inner product spaces are not required, but help the intuition. -- ---------------------------------------------------------------------------- Dr Russell Standish Phone 0425 253119 (mobile) Principal, High Performance Coders [email protected] http://www.hpcoders.com.au ---------------------------------------------------------------------------- -- 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 visit https://groups.google.com/d/msgid/everything-list/ee9f28de-4a2c-4e79-85a6-aef985bb6da3n%40googlegroups.com.
