Re: counter example in analysis

In article <45520E09.1090606@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

On 11/8/2006 7:44 AM, Virgil wrote:
In article <MPG.1fbb250613bdc08d98987a@xxxxxxxxxxxx>,
David Marcus <DavidMarcus@xxxxxxxxxxxxxx> wrote:

Eckard Blumschein wrote:
The genuine continuum
cannot be resolved into points no matter how many times one zooms.

The set of reals cannot be so "resolved", but then neither can the set
of rationals, nor any other densely ordered set.

You seem to be close to the truth. I see it a question of the point of
view. The continuum of reals cannot be resolved into single points
because a resolved IR would be identical with (Q.

As Q cannot be resolved into single points isolated from their neighbors
any more than R can, Eckard is delusional again.

Rationals are always a
set of single numbers.

Any set is a set of single members, if one ignores the ordering. The
rationals are just as dense as the reals when one contemplates their
order properties.

The only difference between them is their completeness properties.
A set of reals bounded above must have a real number least upper bound.
A set of rationals bounded above need not have a rational least upper
bound.

Reals always constitute an unresolvable tea-like
continuum of uncountable fictitious elements.

So according to Eckard's dictionary "real" = "fictitious".


The latter only exist in
imagination, i.e. if one takes the point of view of higher abstraction.

That is where all numbers reside.

The whole point of such density of ordering is that each interval within
the set looks exactly the same regardless of the lengths of the
intervals.

This is correct for rationals. For reals, the points are thought
actually dense with absolutely no intervals in between the fictitious
elements.

And where does one find "intervals" between the "rational" elements.
Density guarantees that there are no such things.

Eckard must really have conniption fits with Robinson's non-standard
reals which have infinitely many infinitesimals between any two reals.


So
the definition does not really refer to actual infinity, at least not
immediately.

On the contrary, only completed infinities of sets can have dense
orderings.

On the intuitive level we usually tend to agree.

I do not agree with Eckard on any level.

Here I just took the
reasoning by Cantor literally. While he considers infinity like
something solid, stationary, a priori given from the very beginning, he
explains "dense" quasi like the possibility to provide as many plugs as
you like.
You are quite right, genuine dense means an actually infinite amount of
"plugs". It is a fiction belonging to the also fictitious "set" IR.

"Dense" applies just as genuinely to the set of rationals as to the set
of reals. The distinction between rationals and reals cannot be a
property that they both have. The only way to distinguish between them
by their order properties is in terms of completeness.
.

Relevant Pages

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