Most different vectors are different lengths.
Brent
On 9/4/2024 5:08 AM, Alan Grayson wrote:
What you write seems correct but doesn't address the issue I raised;
namely, that the metric tensor is defined on pairs of vectors in the
vector space in the tangent plane of the spacetime manifold, and
yields *different *real values for most different pairs. So, it seems
that the metric tensor FIELD is*NOT well defined*. AG
On Wednesday, September 4, 2024 at 6:02:17 AM UTC-6 John Clark wrote:
On Tue, Sep 3, 2024 at 6:44 PM Alan Grayson <[email protected]>
wrote:
/> I fail to see how your comments relate to the possibly
ambiguous concept of the latter. The metric tensor field seems
ambiguously defined./
*A N dimensional space is composed of an uncountable number of
real numbers but it can be unambiguously defined by just N
countable rational numbers, you can pair them up one to one. This
is possible because there is only a countably infinite number of
COMPUTABLE real numbers, the same rank of infinity as the rational
numbers. So you can in effect give a rational number name to every
real number you are able to find on the number line. You can do
this even for a number such as π which is not only irrational,
it's transcendental, because it is also computable. You can use an
infinite series to get arbitrarily close to π. *
*The vast majority of numbers on the number line are NOT
computable (and have no name) but that's not really a problem
despite the fact that the vast majority of numbers on the number
line are NOT computable because, except for Chaitin's Omega
Number, every number that a mathematician has ever heard of is a
computable number. Computable numbers can have names, uncomputable
numbers can not.*
John K Clark See what's on my new list at Extropolis
<https://groups.google.com/g/extropolis>
und
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