Invisible <voi### [at] devnull> wrote:
> The unit square contains an infinite number of points. But is it 
> countably or uncountably infinite?

  I don't really understand why you are asking that. If there are uncountably
many reals in the range [0,1], why wouldn't there likewise be uncountably
many coordinates inside a unit square?

  The more interesting question is if those two sets are of equal size.
If I understand correctly, the set of reals in the range [0,1] has the
same cardinality as the set of points inside a unit square, hence there
exists a one-to-one mapping between them, hence they have the same size.
(Defining the "size" of an infinite set is a bit complicated of a subject,
but an interesting one, if you are in any way into math.)

  In fact, likewise they are equal to the set of points inside a unit cube,
the set of points inside a 4-dimensional unit hypercube, and so on. Even if
you had a multi-dimensional unit cube of countably infinite dimensions, the
set of points would still be the same size as the reals in the range [0,1].

  Now, if the unit cube had uncountably infinite dimensions, the question
becomes more complicated. The cardinality of this set is larger than the
set of reals, but is there a one-to-one mapping between the set of reals
and the set of points inside this uncountably-infinite-dimensional cube?

  I don't fully understand the generalized continuum hypothesis to such
extent as to be certain, but I think that whether such a mapping exists
or not cannot be either proven nor disproven using "regular" set theory
math (for a definition of "regular" that goes well beyond my area of math
knowledge).

-- 
                                                          - Warp

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