Re: Cantor Confusion

In article <1162559784.572072.34420@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

David Marcus schrieb:

mueckenh@xxxxxxxxxxxxxxxxx wrote:
*** T. Winter schrieb:
In article <1162405520.008395.100850@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
mueckenh@xxxxxxxxxxxxxxxxx writes:
> *** T. Winter schrieb:

> > Where in the above quote is 1/oo defined?
>
> It is defined that omega (which Cantor used later instead of oo) is
> a
> number larger than any natural number n. Omega is the limit ordinal
> number. Therefore 1/omega must be a number smaller than every
> fraction
> 1/n.

Why? As long as 1/omega is not defined you can not talk about it. You
simply assume that 1/omega is a number. But that is not the case, it
is not defined.

If omega is a number > n, then 1/omega is a number < 1/n. For all n e
N

What do you mean by "number"? Normally, "omega" denotes a certain
ordinal. Division is not normally defined for ordinals. If you want it
to be defined, then you have to define it. So, using a plausible
interpretation of your words "number" and "omega", your statement is
false because 1/omega is not an ordinal (since it is not defined).

By "number" I understand those mathematical objects which can be
compared by size or magnitude, i.e., which observe trichotomy. I know
that division is not normally defined for transfinite ordinals, but
with omega > n we can define 1/omega < 1/n for n e N as some kind of
abbreviation.

What does Wm allege it abbreviates?

Don't forget, most things come into being by intuition, not by
formalization.

But those things which cannot be formalized tend to go as fast as they
come.

It is by logic we prove, it is by intuition that we invent. (Henry
Poincaré)

But most of what gets invented ends up, as it deserves to, on the scrap
heap. So it what intuition invents cannot be proved by logic, the scrap
heap is its logical resting place.

When Cantor introduced omega, there was no formal definition.

But logic, in that case, provided one.
.

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