Re: Calculus XOR Probability
- From: Virgil <vmhjr2@xxxxxxxxxxx>
- Date: Fri, 05 May 2006 22:05:30 -0600
In article <MPG.1ec544632fc3dec98acc2@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:
Distance and length are real numbers.
Is lim n->oo {C_n} a real number, or is it a set of points?
It's a sequential set of points, that is a line of some sort, with a real
measure called length.
No, it's not.
{C_n} is a /sequence/ of /sets of points/.
C_n for n in N is a sequence of sequential sets of elements, which can either
be the corners of the steps, or the line segments which connect them.
If these C_n's are the staircases, each is an open polygon in the plane,
all of the same end points, the same length and all, having successive
segments forming right angles with each other. The limit as n increases
unboundedly is the line segment between those original endpoints.
If and only if one assumes TO's "infinite inductions principle" is valid
will the length of the limit equal the length of any one of the
staircases.
Ergo, TO's "infinite inductions principle" is disproved.
I observe the following, which you are free to prove me wrong about:
* dist(p,q) = 0 if, and only if, p = q.
* dist(p,q) = dist(q,p) for all pairs p, q.
* For all pairs p,q, and r, dist(p,q)+dist(q,r) >= dist(p,r).
Thus, the function "dist" is a (toplogical) metric by the usual
definition: it satisfies the three conditions above. That makes R^2,
with reference to the function dist, a "metric space" as defined by the
metric "dist".
(One could define other functions which would also be a metric on R^2.
By "the usual metric" in R^n for any n, I mean that the dist function
we want to use is the same as euclidean distance in R^n between the two
points p and q).
Okay, but what happens in the limit, in your estimation? When the distances
between successive points become less than any real number, you seem to think
that the distance is zero or something. But then, how do you account for the
length of the line? How do those zero lengths sum to the overall length of
the
line?
The limit of the sum of distances as those distances goes towards zero,
under suitable conditions, becomes a definite integral, as
mathematicians all know.
If you simply want to forget about infinitesimal line segments, then
you
really can't handle this object. This works well for measuring the diagonal,
but not for summing infinitesimal segments, such as the staircase in the
limit,
without some notion of infinitesmals.
It is a well understood limit process to find the length of a curve,
provided there is an unambiguous limit. There one such value is for each
staircase between two given points, and a different one for the limit
curve as the number of steps increases unboundedly.
.If you're interested in mathematics, then you need to /learn/ some
mathematics; because what you are doing now is not mathematics, it is
some other thing.
Since you are posting in sci.math, I assume you are interested in
learning mathematics. If you aren't, why /are/ you posting here?
Cheers - Chas
- References:
- Re: Calculus XOR Probability
- From: cbrown
- Re: Calculus XOR Probability
- From: Tony Orlow
- Re: Calculus XOR Probability
- From: Tony Orlow
- Re: Calculus XOR Probability
- From: cbrown
- Re: Calculus XOR Probability
- From: Tony Orlow
- Re: Calculus XOR Probability
- From: cbrown
- Re: Calculus XOR Probability
- From: Tony Orlow
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