On 13/07/2013 02:50 PM, Warp wrote:
> 1) Does a unit square contain the same amount of points as a unit line?
> (We are talking about real numbers here.)

Weirdly, yes.

(As an aside, if the answer were no, the rest of the questions all 
become in applicable, which kind of gives the game away...)

> 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.

Given the 2D coordinates of a point on the unit square, you can 
interleave their decimal digits, which always yields a unique point on 
the unit line. For example,

   0.3425
   0.2183 -> 0. 32 41 28 53

> 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?

This amounts to interleaving N sequences of digits rather than just two.

> 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?

Now I'm lost. I can neither prove nor refute that the cardinalities match.

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