Le_Forgeron <jgr### [at] freefr> wrote:
> > 5) And the logical extreme: Does an uncountably-infinite-dimensional
> > unit cube contain the same amount of points as a unit line? Explain why.
> > (Also explain how the number of dimensions can be uncountably infinite.
> > That seems to defy the definition of "dimension".)
> > 
> No.
> Dimension can be uncountably infinite if it is not mappable to the set
> of natural number. Such case can be the set of real number in [0,1),
> (see Cantor's diagonal).
> The number of dimension can be uncountably infinite as soon as the
> number of dimension is no more restricted to a value mappable to the set
> of natural integer. Fractal object have such dimension.

Note that this might be a bit of a trick question. It's talking about
an "uncountably-infinite-dimensional *unit cube*". As you say,
dimensionality being uncountably infinite requires non-integral dimensions.
Thus answering the question correctly would first require demonstrating
whether a non-integral-dimensional unit cube can exist. (I think it can't,
because it goes contrary to the definition of "cube".)

Of course for the sake of the question we can loosen up the requirement
of the object being a unit cube and simply say that it's an arbitrary
object of a unit "volume".

-- 
                                                          - Warp

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