Re: Is continuum completely filled up?

In article <$EmMO1IYI7tFFwwj@xxxxxxxxxxxxxxxxxxxxxxxxxx>, Andy Smith writes:
David Marcus <DavidMarcus@xxxxxxxxxxxxxx> writes

Is this a serious question? Or, are you trolling? How can you ask a
question about "successive points" and then add "(there can be no such
thing)"? And, what does "continuous" mean in this context?

Well, yes it was a serious question. This thread is in the context of
"is continuum completely filled up". So I had previously suggested that
you could possibly have holes, depending on how you "construct" the
reals

If you construct the reals using Cauchy sequences, it can be *proven*
(and is in elementary Real Analysis) that the reals are complete.

e.g. the 2D fractal curve filling all space,

I've never seen a construction of the reals based upon a fractal curve.
I would guess that it would be difficult to even define a "fractal
curve" without already having a construction of the reals. If you
can construct, in a non-circular way, the reals from a fractal curve,
more power to you.

But, just saying "if we construct X in manner Y, X would have problem
Z" is only a valid argument against somebody who has proposed using
method Y to construct X. If people have constructed X using method C,
a flaw in Y is a straw man.

if we are concerned about holes amongst the reals,

We're not, because we've proven that there aren't any, or a least
read one or more proofs. For an introduction, the following pages:
<http://en.wikipedia.org/wiki/Real_number>
<http://planetmath.org/encyclopedia/MathbbR.html>
<http://www.cartage.org.lb/en/themes/sciences/Mathematics/calculus/realnumbers/complete/complete.htm>
seem reasonable, although I'll admit that I've only skimmed them.

and given that no two real points are adjacent, but that continuity is
assured by recursive procrastination

Well, no, it's assured by the convergence of Cauchy sequences within
the reals. I'd strongly recommend reading up on the basics.

perfectly reasonable to ask how much one can increase the parameter "a"
by for the linear interpolation.

You can make "a" as large as you want -- there's no upper bound to
the reals.

Clearly you can't increase it by "just
enough" to ensure that successive points touch

There are no successive points. Period.

Anyway, even if from your perspective it is a dumb/incomprehensible
post, no I'm not trolling.

The way to convince us of that is to go read some of the references,
such as the Wikipedia article.

--
Michael F. Stemper
#include <Standard_Disclaimer>
91.2% of all statistics are made up by the person quoting them.

.

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