Re: Cantor Confusion

In article <1162135345.044588.103450@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx wrote:

*** 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.

It is. however, quite sufficient that the constructed number be
different from the nth listed number by at least 1/10^n


How many different representatives can be chosen in Cantor's list for
one equivalence class?

One is enough.


In Cantor's list there are omega irrational numbers.

It isn't Cantor's list, it is any list of reals presented to Cantor.

Cantor uses it. Otherwise everything was finite.

Only in WM's world is omega itself needed to make things not finite.

Everywhere else, one can get by with using only members of omega.

> > > You will need it in order to construct a real number and its
> > > decimal
> > > representation for a Cantor list.
> >
> > No. By the theory, each decimal number is a representative of an
> > equivalence class of sequences of rational numbers. By the
> > construction
> > we get another decimal number that is also a representative of an
> > equivalence class of sequences of rational numbers. No omega is
> > needed.
>
> Why then do you think omega is needed at all in mathematics?

Because it comes in handy on many occasions. What *is* needed is the
axiom of infinity, because that guarantees that you can even talk about
infinite sequences.

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

The axiom of infinity merely says at least one set, and in no way
requires only one set satisfying its properties. In fact, if any one
such set exists, there must be many others as well.

(And ultimately about limits.) Without that axiom
there is no way to prove that infinite sets do or do not exist.

Correct. And with this axiom the first infinite entity which is proven
is omega. And then the digits of any irrational are enumerated. Their
number is omega.

Their number is the cardinality of omega. "WMueckenheim" really should
learn the difference between ordinal numbers and their cardinality.


I would not know how to do that. Perhaps that is possible, but in that
case you would need to reformulate the limit concept.

Why then do you try to dispute that omega does play a role in
constructing he reals? The axiom of infinity essentially says: There is
the set omega.

It says both more and less than that.

> Omega is introduced only by the axiom of infinity.

No. It is *defined* using properties obtained through the use of that
axiom. In some fields of mathematics you do not need omega at all, but
only the axiom of infinity (analysis, and I think also algebra, number
theory, and a host of other fields).

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

NO! Those properties are held by every limit ordinal, of which there are
many besides omega.
.

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