Re: Calculus XOR Probability


Tony Orlow wrote:
Matt Gutting said:
Tony Orlow 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 soemthing that doesn't exist?

No, and I'm not. It's not true that "n approaches infinity"; n increases without
bound. And one can certainly take a limit as n increases without bound.

Right, but you can't say what the curve IS in the limit without considering
having REACHED the limit.

You have no idea what the mathematical definition of limit is. The
entire idea of the limit is to formalise the notion of "increasing
without bound" without having to talk about "reaching infinity".

You have a "taxicab" distance of 2? It doesn't matter WHAT
rectilinear approaching path you take, it'll always be 2. So, if you think the
limit of the staircase DOESN'T have a length of 2, it's not a taxicab distance,
and the object is no longer a staircase.
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?

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?
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.

See my reply above.


The staircase approaches the diagonal, location-wise, but in the end, it's not
the same object, as I've demonstrated.

The sequence of staircases "approaches the diagonal" arbirtarily
closely. Then the limit of the sequences of staircases is the same
object as the diagonal. That's what "the limit" means....

What does "in the end" mean, here?

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. while the treads and
risers become infinitesimal, their direction never changes, and never
approaches the direction of the diagonal.

You haven't defined "infinitesimal" to anyone's satisfaction, certainly not
to mine.

Any finite divided by any infinite yields an infinitesimal. If you divide the
unit interval by n, as n->oo, the subintervals have lengths approaching 0. For
any specific, non-absolute infinity, you have an infinitesimal value. As the
number increases without bound, the subintervals shrink with a bound of 0, but
never ultimately reach that bound, as n never ultimately reaches absolute oo.
But, perhaps you just aren't partial to infinitesimals.

What is a "specific infinity" and an "absolute infinity"? What does
"ultimately reach" and "absolute oo" mean?

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.

See?

No; you're assuming that the limit is some construct involving infinitesimals.

I am assuming that when n is a specific infinity, 1/n is a specific
infinitesimal, and that's part of the problem we're discussing.

Still, nothing to do with limits. "Specific infinites" are precisely
what the idea of limit is trying to avoid. The "specific infinites" in
your theory are just meaningless labels.

Q What is "BigUn"?
A The number of reals in the unit interval!

Q What number is that, then? What *type* of number? What are its
proeprties?
A BigUn. An infinite number. I'll tell you when I have it all worked
out.


<snip>
To an extent. I was indexing curves by their position in the sequence, which
was coincidentally the number of segments; so that agrees with what you're
saying. But I'm still unclear about what you mean by "we have the 'curve
in the limit'." Is the "curve in the limit" one of the curves in your
sequence of curves?

Sure, if we consider n from 1 through Big'un, we can sonsider the value at
Big'un to be the value in the limit.

So your concept of "limit of a sequence" is just stopping after an
arbitary amount of steps through the sequence? Can a sequence have more
than one limit, depending on if we "consider n from 1 through Big'un"
or consider something else?

If so, at what position can it be found?

Big'un.

And what position is that? After how many steps do you reach it? How do
you know? How do you make sure you haven't gotten to BigUn-1, by
mistake?

Is BigUn a "specific infinity"? Can you give it a definition, some
properties? Can you distinguish it from BigUn-1?

<snip>

--
mike.

.

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