Le 13/07/2013 15:50, Warp nous fit lire :
> 1) Does a unit square contain the same amount of points as a unit line?
> (We are talking about real numbers here.)
> 
Yes. It must be, from Cantor, something bigger than Aleph-0. (and is not
countable)


> 2) If yes, that means there has to be a 1-to-1 mapping between those
> points. Give a function that expresses such a mapping.

IIRC, Peano curve would do. Expressing the position of any point using
(x,y) can be replaced with the ratio (between 0 and 1) over the Peano
curve of the length to reach the position over the total length of the
curve (and the precision of the ratio number can be used to infer the
order of the peano curve, or you can use any very big order of Peano
curve as it suits you)

You can also consider Wunderlich curves. Same approach. All real numbers
in [0,1) maps to the points on the unit square.

> 
> 3) If the answer to the first question is yes, then it follows that
> the amount of points inside a unit cube is also the same as the amount
> of points on a unit line. The same for a four-dimensional hypercube,
> and so on. Can you give a generic function that gives a 1-to-1 mapping
> between a unit line and an n-dimensional unit cube?

Be careful, using infinite and common sense might yields some surprise.
A basic approach would be to convert the first two coordinates (both in
[0,1) ) into a single coordinate using either Peano or Wunderlich, and
then repeat.

 X,Y -> a
 X,Y,Z -> a,Z -> b
 X,Y,Z,T -> a,Z,T -> b,T -> c

 C0,C1,C2,... Cn -> (C0,C1)->p0,C2,... Cn

> 
> 4) So the next question is: Does a countably-infinite-dimensional unit
> cube contain the same amount of points as a unit line? If yes, can you
> give a 1-to-1 mapping between them?

Yes, apply the answer of point 3, once the dimension value has been
mapped (because countably-infinite) to the set of natural number , in
order to have the number of original coordinates.

The mapping (of dimension) is obvious, because it is the definition of
the countably-infinite. ("obvious" as in math, after 10 to 2000 pages of
demonstration and 40 minutes or 500 years of intense thinking)

> 
> 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.

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