Le 13/07/2013 18:55, Warp nous fit lire :
> 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".)

Nope, the question is similar to asking: can a white crow swim ?
The answer does not have to prove the existence of a white crow. But
using the properties of such white crow, answer the question of the
capability to swim or not.

Nevertheless, my answer is bit out of scope, regarding the question: it
just asserts that the previously used method of mapping to a single
value cannot be used with a non-integral-dimensional unit-cube. Maybe
there is a different method for mapping even transcendental numbers.

> 
> 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".
> 
We "just" need to define the notion of distance in
non-integral-dimension, and from that just cut any cube out of such
setting. But that's not the question. (cutting a sphere is easier, but
cube are just sphere too)

If we immerse the non-integral-dimension cube into an integral-dimension
space (of higher dimension), it is obvious that we can map. But then the
question becomes: is there a way to immerse a non-integral-dimension
into an higher-integral dimension. (and then, yes, does such white crow
exists ?)

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