Re: counter example in analysis

On 11/6/2006 4:49 PM, Dave L. Renfro wrote:
Robert Israel wrote:

f(x,y) = x y/(x^2 + y^2) for (x,y) <> (0,0),
0 at (0,0)

Eckard Blumschein wrote (in part):

Isn't your examle rather an illustration for questionable
singularities in IR? With rational numbers, there is no doubt:
(0,0) is an exactly accessible address. However, things may
be different with real numbers at least if we understand
them like they were designed for Cantor's DA2.

I don't know what you're talking about, but Robert Israel's
example is a standard example that is given most elementary
calculus texts. I'd be interested in what your arguments
against this example were when you took beginning calculus,
and what your teacher said in response.

My arguments are quite simple and hopefully compelling:
Let's consider like an example the interval of real "numbers" between 0
and 1.
As long it contains just arbitrarily many numbers, these numbers can be
different from each other and may be called rational numbers. There is
no largest amount of them. In this case, I may and I have to distinguish
between open and closed intervals.

Between two munbers, there is always a potentially infinite sauce-like
continuum of locations. Nonetheless, according to Dedekind's wording,
any countable amount of rational numbers is overall dense.
I just envison two possibilities to cover the whole line with points.

The first one is motion. Remember the fluxus of indivisibles for
instance tought by Pythagoreans, Cavalieri, Descartes, Torricelli and
Newton.

The second one is the very useful but in the eyes of not very
open-minded experts like Berkeley nonsense fiction of perfectly much
rather than many fictitious real numbers. Berkeley was mocking: neither
finite quantities, nor quantities infinitely small, nor yet nothing. May
we call them the ghosts of departed quantities. I am tempted to answer
yes. Regarding the attribute fictitious I refer to Leibniz. Having
looked into the valuable botch by Betsch, I feircly maintain that reals
are just fictitious numbers and this distinction does not have anything
to do with physical constraints. It belongs to the fundamental
difference between rational and irrational numbers.

I imagine the rational numbers embedded like sugar into continuous fluid
of irrational numbers. The continuum of irrational numbers will not at
all be changed by addition or removal of the rationals. Infinity is hard
to imagine. Spinoza defined it the quality of being neither enlargable
nor exhaustable. The understanding of this principle seemingly vanished
by and by until Dedekind and Cantor were encouraged by
E. Heine to come up with a cheeky mutilation of the notion infinity.
Cantor perhaps suffered on a regular basis on scruples. So he called
infinity a gulf.
On the other hand he impressed the world with DA2: There are infinitely
many rationals. So there must be more than infinitely many reals, he
argued. Having looked in vain for tangible evidence in support of
transfinite numbers and set theory, I will only mention that Cantors
definition of infinity was admitted to be untennable by Fraenkel.

We do not need the overdue clarification how to get out of set theory.
We may immediately consider how to consider the fictitious "set" of real
numbers. The reals must be selfcontradictory: Irrational "numbers" are
defined as fictitious solutions to problems/tasks that cannot be
numerically resolved except with an fictitious perfectly infinite amount
of numerals.

Does any single fictitious element out of the tea of reals matter for
the entity? Devided by oo it has effectively no influence at all.
Therefore, IR, IR+ and IR- may all have their own nil. One may perform a
symmetrical cut within IR but not within (Q. Buridan's donkey cannot
suffer starvation because its head can not at all exactly be positioned
in the middle.

Eckard Blumschein




So far I did not yet face any

.

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