Re: Cantor Confusion

In article <1162299756.756951.78990@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:

*** T. Winter schrieb:

> For the diagonal number of Cantor's list it is nnot sufficient to come
> arbitrarily close to a number which is different from any list number.

Sorry, but numbers are fixed and not variable. The sentence "number ...
to come arbitrarily close" is nonsense. Numbers do not come arbitrarily
close to each other, it is sequences that can come arbitrarily close to
each other.

Irrational numbers have no last digit. Therefore, with a sequence of
digits like the diagonal number is, one can never have a completed
number but only come as close as possible to any number --- or avoid to
do so.

Yes. So what? The sequence comes as close as one wishes to some other
sequence. What is the problem with that?

> > Representatives of the equivalence classes that actually are the real
> > numbers.
>
> How many different representatives can be chosen in Cantor's list for
> one equivalence class?

As many as you want.

Not so. Of course we talk about a fixed base like 10.

Ah. In that case one or two, depending on the number involved. But
the diagonal obviously depends on the actual representative chosen.

> Omega is but a convenient name of the set which exists according to the
> axiom of infinity.

'the set' -> 'a set'.

OK. The smallest infinite set.
> The axiom of infinity states that there is a set with such and such
> properties... And this set is omega.

I think this is NBG.

It is ZF too.

Not implicitly. It depends on the model we are using.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.

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