On 7/13/2013 6:50 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.)

Taking "same amount of points" to mean "the two sets of points have the 
same cardinality", then yes.


> 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.

An injective mapping suffices to prove #1 (since the other direction is 
trivial), and interleaving the digits achieves this.  A 1-to-1 mapping 
is trickier due to annoying edge cases with multiple decimal 
representations of the same number.  I don't see a (constructive) way to 
do a true 1-to-1 mapping off of the top of my head.


> 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?

My answer is identical to #2.


> 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?

Assuming you take the "obvious" definition of an "inifinite dimensional 
unit cube" to actually be a unit cube (which I think is natural to do), 
then yes.  Again, only an injective mapping is obvious to me.  This can 
be done with a triangular interleaving type pattern.


> 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".)

As you've stated it, I believe the answer to this depends on the 
continuum hypothesis.

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