Re: Two results of set geometry

WM wrote:
On 16 Sep., 18:24, Tony Orlow <t...@xxxxxxxxxxxxx> wrote:

I never noticed it, although I walked, during the last weeks, every
day for several hours through the Harz forrests, a landscape which
also Cantor loved very much.
That sounds nice. I've been doing some gorge walking lately, here in
Ithaca. Nature is good inspiration. Water is medicine. When two ripples
on a pond meet at a point, and that point becomes two, how fast are they
moving from each other, at that moment?

Or take relativistic scissors. Or take the relation on R: x = 0 for x
=< 0 and x = 1 for x >= 0, how fast is it raising at x = 0? Do you
want to hear infinitely fast? Of course we may call it so. You can
also say: instantaneously. But there is no infinite set. That is my
point. A set consist of elements which one must be able to
distinguished on demand.

Do you disagree that, for any rational number, it can be distinguished from pi, as being either greater or less than pi?


Where and what are these points? Hegel argued that the front side (of
a plane) must have a rear side and a front side itself, while this
front side again must have a rear side and a fron side, and so on.
This reminds me of people saying that all the points are "there".
Hmmmm... That may remind you of irrationals, but I think it's a
different kind of argument. I don't know what Hegel was trying to prove,
but there exist numbers on the real line, like e and pi, which cannot be
expressed as any finite rational fraction. Those are the numbers that
require an infinite number of subdivisions, and they far outnumber the
finitely subdivided subsegments. That's how I see it.

Therefore I do not believe that they exist on the real line - they are
merely ideas. One cannot prove the existence of the point on the real
line. Therefore Cantor stated, what is known as his axiom, that there
is a point for pi etc. I do not accept it, because it requires a
finished infinity of operations or digits. Who does not believe in
that should not believe in pi. Cantor believed in that. He was
consistent. No modern mathematician believes that. They all are
inconsistent.

All numbers are ideas, but there is a difference between the point on the real line and the digital representation of it. No completed digital representation, in any natural base, exists for pi. That doesn't mean the point doesn't exist on the real line. It can be distinguished from every rational number, in the general quantitative order of the line.

Hint: The irrationals replace all the redundant rational values in the
infinite matrix of fractions. Those far outnumber the unique values.

Of course, that would apply to rationals as well. and so speaks to
density on the real line. That is required by finite range. Still, one
can have an infinite number of naturals, with naturals of infinite
value
Then, by definition (and by the common perception), these numbers are
not naturals.
They are certainly not standard naturals, that's true. They are members
of the set of set sizes, or counts.

Nobody counts so far. If I count to some number, then, by definition,
this number is a normal natural. My counting to it proves that.

Yes, but if one assigns a count to, say, the unit interval of reals, then relative counts can be expressed for other infinite sets of numbers. If I count ten unit intervals, then I've counted ten of these infinite units. If we consider points to be atomic elements of space, then any space contains some integral number of points. No finite non-zero amount of space contains only a finite number of points.

He says that there are no actually infinitely small numbers, and that
we don't need them because of the work of Cauchy and other famaous
French mathematicians. (He means, we need only finite epsilons and
deltas.)
Certainly, the method of limits using only arbitrarily small but finite
differences has supplanted Newton's and Leibniz's original uses of
infinitesimals. They are not "necessary", necessarily. And, there is
objection to "inconsistent" use of them, where they are significant at
one point in the calculation, and discarded later.

For instance why is x + dx = x in case x = 0?

Not sure of the context here, but the dx gets thrown away in the end, either way.


However, the notion
of the infinitesimal had been rigorized and, I believe, that it still
has plenty of credibility, and that acceptance of actual infinity leads
inexorably to acceptance of the infinitesimal. Just my two cents' worth.

Not for Cantor.

Regards, WM


No, Cantor doesn't dig infinitesimals.

Peace,

Tony
.

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