On Sat, 13 Jul 2013 15:50:42 +0200, Warp <war### [at] tagpovrayorg> wrote:

> 1) Does a unit square contain the same amount of points as a unit line?
> (We are talking about real numbers here.)
>
> 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.
>
> 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?
>
> 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?
>
> 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".)
>

I don't know.

Maybe I should start with something simpler.

Infinity + Infinity = ?

Is there any other answer than 2(Infinity)?

If so please explain.

-Unlike Warp I don't know the answer and really want to know :)

-- 
-Nekar Xenos-

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