On 7/13/2013 8:50 AM, Warp wrote:
> 1) Does a unit square contain the same amount of points as a unit line?
> (We are talking about real numbers here.)

Saying that they have the same number of points is not a meaningful 
statement.  However, we can say that they have the same countability.

Consider any ordered pair (u,v) in the unit square.  These two values 
can be presented as

0.u_0 u_1 u_2 u_3 u_4 u_5 ...

and

0.v_0 v_1 v_2 v_3 v_4 v_5 ...

Where the different u_n and v_n are digits of whatever radix you care to 
select.

 From these two numbers you can construct a third number thusly:

0. u_0 v_0 u_1 v_1 u_2 v_2 u_3 v_3 ...

Essentially you can interleave the digits of u and v to produce a third 
number w that falls in the unit line.

> 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.

Done.

> 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?

I'm sure you can extrapolate from the above example to any higher dimension.

> 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, as you can see from above.  The unit line, unit square, unit cube, 
unit hypercube, etc., all have the same countability.

> 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".)

The hypercube of infinite dimension may have a higher countability than 
any finite-dimensioned cube, but it is certainly no lower in countability.

Regards,
John

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