Re: Cantor Confusion

In article <45793B7C.4050105@xxxxxxxxxxxxxxxxxxx>,
Eckard Blumschein <blumschein@xxxxxxxxxxxxxxxxxxx> wrote:

On 12/8/2006 1:54 AM, *** T. Winter wrote:
In article <45781DC1.1000804@xxxxxxxxxxxxxxxxxxx> Eckard Blumschein
<blumschein@xxxxxxxxxxxxxxxxxxx> writes:
> On 12/7/2006 1:54 AM, *** T. Winter wrote:
...
> > Oh, well. In Bourbaki's mathematics R+ and R- both contain 0. So you
> > are a follower of Bourbaki after all? But of course the 0's are the
> > same. If they were different you would have quite a few problems with
> > limits and continuity.
>
> According to my reasoning, any really real number is not unique but must
> rather be void because even the tiniest interval is thought to contain
> indefinitely not just many rational numbers but indefinitely much of
> real numbers.

But what you state here for real numbers is also valid for rational
numbers.
So you make not clear why in one case they are "void" (whatever that may
mean) and in the other case they are not void.

I beg your pardon. Though the matter is clear to me, I fear it is
difficult to explain because mathematicians are not used to think in
terms of notions with higher abstraction.

On the contrary, mathematicians are used to thinking on much higher
levels of abstraction than most non-mathematicians. It is the nature of
mathematics to abstract.




So meanwhile every dull mathematician tells and is
even teaching that all numbers are fictitious.

Wrong. mathematicians never call numbers fictitious. It is only dull
non-mathematicians who insist on trying to teach their grandmothers to
suck eggs.

While the imaginary numbers are obviously different from the ordinary
numbers, the corresponding distinction between rationals and reals is
more subtle.

Rationals are ratios of integers, non-rational reals are not ratios of
integers. That should not be too subtle even for EB.


Perhaps it is most helpful to declare the reals just
fictions, while the rationals, including naturals and integers, are
genuine numbers.

It is most unhelpful to misuse words whose common meanings tend to
mislead one about their technical meanings.

The difference between rationals and reals corresponds to the difference
between potentially infinite and perfectly infinite.

Since in such set theories as ZF or NBG or NF there do not exist any
such things as potentially infinite sets but there do exist infinite
sets, the distinction is irrelevant in those set theories. And in those
theories each real is a set just as each rational is a set and each
natural is a set.

If EB wishes to produce an axiomatic system which distinguishes between
potential and actual, let's see him do it.

But absent such a system, there is nowhere that such a distinction
exists.




Infinity is in some sense the opposite of being infinite.

Sanity is clearly the opposite of being EB.
.

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