Re: Calculus XOR Probability

In article <MPG.1ec519114c0e319098acc0@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Virgil said:
In article <MPG.1ec3d7695f98b59c98acb6@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

cbrown@xxxxxxxxxxxxxxxxx said:

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.

What does TO mean by a "sequential" set of points.

The usual meaning of a "sequence" is a set ordered in such a way that
all but possibly one member has an immediate successor, all but possibly
one has an immediate predecessor, and that any lessor member is linked
to any greater through a finite set of successors.

Since TO cannot impose this structure on the points of a line, he must
mean something else.

I mean everything above except for the limitation of finiteness on the number
of successors between any two points.


This cannot happen with any subset of the reals.

If one has an increasing sequence of reals each of which is smaller than
all the members of a decreasing sequence of reals, then there is ALWAYS
at least one real strictly between the two sequences which is not a
member of either of them, thus not connected to either of them by ANY
sequence of successors, finite or otherwise.

And thus neither sequence is connected by successorship to the other
either.


So the reals provably do not behave the way TO says they must.



So why do you claim that I am using the limit to measure distance or
length? I am using the limit to take the limit of a sequence of curves,
nothing more.

To measure distance or length of a set of points in R^2 with the usual
metric, I use the usual approach, just like everyone else.

That's good, because this approach doesn't work.

It has worked perfectly for everyone for thousands of years. Why should
it stop working now?

When you prove the length in the limit is 2 instead of sqrt(2),
that's an indication that the object in the limit is NOT exactly the
same as the diagonal line

No one has proved that without referring to TO's principle of "infinite
induction" so that it must be TO's principle which is at fault.


As he does not prove any such thing, TO is way off base.

That was the concluson Chas logically drew, erroneously or not.

Only by appealing to TO;s principle of infinite induction, so it
disproves to" principle of infinite induction by contradiction.


What he does prove is that TO's version of arc length behaves that way,
but that is quite different from any actual version of arc length.


It wasn't MY version of arc length.

If TO insists that a point must have a direction or that what holds for
finite equalities must always hold in the limit, then it IS TO's version.


and the reason for the
discrepancy is easily seen to be due to the angle at which each of the
approximating segments intersects the diagonal it's supposedly measuring.


Angles are irrelevant. For each finite sequence of points along a piece
of an arc, one can find the point-to-point length without any attention
being paid to any angle, but only to the distances between successive
points. If the set of all such polygonal lengths has a least upper
bound, that LUB is the length of the curve.

Yeah, if the segments are parallel at some point to the curve.

Not so! One can do it with polygons approximating staircases, which
polygonal approximations, except for one end bit, are nowhere parallel
to any part of the staircase.


I do not know what sort of measure of a curve TO is talking about which
does not fit that definition. And neither does TO.

There's a lot you don't know, despite your opinion of yourself.

But I admit my ignorance, where TO tries to cover his up.



Do you understand what I'm saying about the elements of the staircase
needing
to be parallel to the diagonal in order for the "usual metric" to work?

We can easily understand what you are saying, but not why you are saying
it, because it isn't true.

If you say that, then you obviously don't understand what I'm saying.

Au contraire!



No, you are misunderstanding me. I reject your example as demonstrating
that
the concept of an inductive proof of equality holding in the infinite
case is
invalid, because the error in your staircase example is easily
explainable in
the way I've been describing.

But since that easy explanation is quite false, its ease does not excuse
its irrelevance.

Your declaration of falseness is false. That handwaving nonsense is pathetic,
Virgil. Really, you can do better than that. Why don't you go measure a sine
wave using vertical elements, and tell me how that works out? Show me one
example where the segments are not parallel to the curve, which actually
works,
or admit you're full of crap.

It is TO who is full of it, as here is the demonstration for the
staircase function under discussion:

Given a staircase of n steps (2*n segments, each at right angles to the
ones at its ends.

The first polygonal bit goes from one end of the staircase to a point
epsilon short of the first stair segment, and from there to epsilon
short of the end of the next segment , and so on.

This polygon has only one piece parallel to any part of the stairs, but,
for small enough epsilon, will have length arbitrarily close to the
actual staircase length.

The LUB of such polygon lengths equals the staircase length.



What is the usual method for measuring the staircase in the
limit? As far as I'm concerned, you proved it had length 2.


As TO is totally unfamiliar with distinguishing between valid and
invalid proofs, his judgement of what constitutes a valid one is just
another of his irrelevancies.

Irrelevancies? Whatever. The proof is valid. The limit is not of a sort which
can be used to measure arc length of the diagonal, however. They are slightly
different objects, one with length 2, and one with length sqrt(2).

Since they are eachmerely sets of points and each point of either is a
member of the other, in what respect are thy different?

Yes! TO's alleged principle of "infinite induction" has been shown to be
invalid by impeccable logic several times, but TO's unreasoning faith
in it persists.


The usual metric for the Cartesian plane is that for points whose (x,y)
coordinates are (a1,b1) and (a2, b2), the distance between them is
sqrt( (a1-a2)^2 + (b1-b2)^2 ).

Similar formulae hold in Cartesian spaces of other dimensions.


Oh, well, that's just the Pythagorean distance formula in Cartesian
coordinates. According to that, the staircase has length 2.

Parallelism is irrelevant, and need not even be defined in the space in
question, so long as it is a metric space.

TO seems in every mathematical question to have an unerring ability to
single out one irrelevant property as being the essential one in his
warped version of mathematics.

The distance between two points does not depend at all on the direction
between them. If it did, then rotating the space in which the points
occur would change the distance between them.


The sky is blue, Virgil, and love is good. (sigh) And puppies are cute.

TO, when cornered on the mathematics being discussed, escapes again into
his fantasy world outside of mathematics!
.

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