On Fri, May 27, 2011 at 07:03:11PM -0600, Jason wrote:
> This is what i get for breaking it down into lay mans terms.
> Maybe this will be more palatable.
> the length of 0-1 = 1
> the length of 0-2 = 2
> to find the number of points in a given length for a given distance
> between those points you take the total length divided by the length of
> the distance between those points
I now understand what you were trying to say, but it's still not quite
the way mathematicians like to think about it. For one, this line of
thinking doesn't distinguish the cardinality of the set of rational
numbers from the cardinality of the set of real numbers.
> however when calculating an infinite number of points in a given length
> you end up dividing by 0 because there can be no distance between each
> of the points. Otherwise you would get a finite number of points between
> the lengths.
Not necessarily. Consider the union of {0} with the interval [1, 2].
There are an infinite number of points between 0 and 2, but there is
also a big gap (between 0 and 1). My counterexample is trivial, but a
big part of measure theory is thinking about various types of "holes" in
infinite sets.
> 1/0 on a positive number line = infinity
> 2/0 on a positive number line also = infinity
> which is what we expect because we stated that we wanted an infinite
> number of points and to compare the quantity of them According to all
> the math 1/0=2/0 thus there are the same number of points between 0
> through 1 as there are through 0 through two.
There are infinitely many rational numbers between 0 and 1, and there
are infinitely many real numbers between 0 and 1, but the cardinality of
the set of reals is greater than the cardinality of the set of
rationals (i.e., the number of points is not the same).
--
Andrew McNabb
http://www.mcnabbs.org/andrew/
PGP Fingerprint: 8A17 B57C 6879 1863 DE55 8012 AB4D 6098 8826 6868
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