On Fri, May 27, 2011 at 7:03 PM, Jason <[email protected]> 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
> 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.
> 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.
Like I mentioned before, set theory cardinality and calculus have
somewhat different concepts of infinity. Actually, 1/0 does not exist
on the standard real number line, but the limit of 1/x as x goes to 0
(from the positive direction) is the placeholder value infinity, which
is added to the extended real number line (along with -infinity) as a
placeholder to deal with this case. In this case, there are only two
infinite values, and they are just abstract 'limit' points on the
number line. There's another extension of the real numbers that loops
the number line so that both the positive and negative directions meet
at a single infinity point, but that distinction isn't relevant to
this discussion.
Anyway, because infinity is just an abstract placeholder in calculus,
and also calculus only deals with the real numbers which we already
know is > aleph0 (and is aleph1 if Cantor's continuum hypothesis is
true, the proof of which relies on the axiom of choice so is not
universally accepted), your argument only tells us that the size of
both sets is > aleph0. Daniel claimed that [0..2] on the real line is
larger (has a greater cardinality) than [0..1], but your argument
doesn't speak to that.
Anyway, after a bit more research, I found that you can use the arctan
and tan functions to map between R and any open interval within R,
which does prove that Daniel was wrong and the two sets he mentioned
were in fact the same size. In fact, since 0..1 is a subset of 0..2,
the elements map with the identify function, and you can map 0..2 back
into 0..1 by dividing each element in the set by 2. Easy!
Now, if you want to know a set with a higher cardinality than R, I
can't help you there.
--Levi
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