Re: Calculus XOR Probability

In article <MPG.1edd186940f90f1198ad14@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Matt Gutting said:
Tony Orlow wrote:
Matt Gutting said:


*My* question is, since you haven't actually defined oo, how can you
tell
whether oo or 2/oo exist?

Because that's the LIMIT. You want to take the limit as n->oo?

Yes, or writing it out without shorthand, I want to take the limit as
n increases without bound.

Well, oo has to
exist, doesn't it?

Not necessarily.

Oh. Then the symbol doesn't necessarily mean anything. Can you take a limit
as
n approaches something that doesn't exist?

One can take some limits as some variable increases without limit.

Lim{x -> oo} is shorthand for
"The limit as x is allowed to increase without bounds"
and the "oo" is no more meaningful out of context that is a letter
outside its word.


That's exactly it. There's no requirement that the limit object be the same
sort
of thing as the members of the sequence.

You don't require it, perhaps, but then again, if you think they are the
same,
then what happened to your arclength measure?

It knows better.


If it's still a staircase, with an
infinite number of infinitesimal stairs, the length IS 2, because that's
the
nature of the staircase. In any case, you're talking about the limit as
n->oo,
so what makes YOU think oo exists?

See above. it is a symbol which no more has the same meaning out of
context that do letters outside of the words they form.

I'm not talking about the limit as n approaches anything, as you seem to
imply
from the way you write "the limit as n->oo". I'm talking about the limit as
n
increases without bound. I don't believe oo exists as a number.

Then you have no business talking about the identity between the staircase
"in
the limit" and the diagonal. If oo doesn't exist, then they never are the
same,
and the whole discussion goes out the window.

Wrong!

One has to look at the definition of the limit.

In this case one has, in mathematics, the formal definition of a
sequence of curves converging to a limit curve which never requires that
anything like oo exists.

And according to this definition, the limit of the sequence of
staircases is the diagonal segment.



Of course, you asked a different question from last time, so I am not
sure you
know WHAT you're asking. The limit of the staircase is a staircase in the
limit.

Can you prove that assertion?

I have demonstrated a concept of limit that shows it.

As that "concept" contradicts mathematical definitions, it is not valid
in mathematics.




The difference between the diagonal and the staircase cannot be
distinguished by location alone. By defining the curve as a sequence of
segments, rather than a set of locations, the difference is quite
detectable,
because the segment definition preserves the notion of direction IN THE
LIMIT.

What is the direction of a point?

If the locations are truly indistinguishable, then the endpoints are
identical,
and the result is a point, not a segment.

Incorrect.

What is the direction fro (x,y) to (x,y)?



Infinitesimal differences are not
equalities.

Infinitesimal differences are not the limit required. If infinitesmal
means greater than zero then we have not reached the limit yet.
.

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