Re: Cantor Confusion


Franziska Neugebauer schrieb:

Please look it up in any modern text book. You will find there that
omega is a cardinal number too.

Anyway, omega is not a /natural/ number.

And a cow is not a horse. Does that eliminate your error of claiming
omega =/= |omega| in modern set theory?

You need not interpret n as a set (though you can do it).

In contemporary set theory almost everything is a set.

In ZFC everything is a set. Not in every set theory.

I don't want to debate about the axioms of ZFC being sets. The point is
that you want to talk about "columns" which neither ZFC nor any other
contemporary set theory is about.

I think that one can introduce new expressions, examples, and
illustrations if they have been made sufficiently clear to a
correspondent of average IQ. The reaction of William confirms that my
ideas are understandable. So your reaction does not concern my writings
but rather your means of reception.

A treatise in which variables ("n") and number symbols ("0", "1",
...) do not refer to sets is not a treatise _on_ set theory but
a treatise of _application_ of set theory, if ever.

Don't mistake set theory with ZF or ZFC.

A theory of _sets_ is not a theory of _columns_.

Experience has shown that practically all notions used in contemporary
mathematics can be defined, and their mathematical properties derived,
in ZFC. In this sense, the axiomatic set theory serves as a
satisfactory foundation for al other branches of mathematics. It can
describe every mathematical notion --- with the exception of what a
column is?

In contemporary set theory it is said that omega is _the_ _set_ of
natural numbers [as Virgil pointed out the _ordered_ set]. The number
(cardinality) of omega is named aleph_0. So it is said that the
number (cardinality) of _the_ _set_ of natural numbers is aleph_0.

You are completely in error. The number (Anzahl) of a set is its
ordinal number.

"Anzahl" is your (or a historical person's) wording not mine. I can't
spot any error in my wording.

That is due to your incompetence.

Sometimes it is necessary to quote. In particular if you are
uninformed but nevertheless refuse to take advice from me.

I don't need any advice from you.

You don't know it. That' s why you would need to learn a lot. Look, you
have learned from me meanwhile, even against your furious opposition,
that omega = |omega| in modern set theory. Doesn't this case make you
wonder whether there are other things which you do not yet know but
which you could learn from me?

My opinion is

A (n e omega & |{0, 1, 2, ..., n}| < |omega|)

Seems an empty opinion. What is the symbol A refers to?

As the diagonal is defined to consist of the ends of terms
of lines, this claim is easy to conradict.

"ends of terms of lines" is your wording not mine.

It is not your wording, because you prefer to veil your
inconsistencies, but it is your opinion.

My opinion is

|(d_nn) n e omega| = |omega|

I know. You say there are less natural numbers than omega, but there
are as many natural numbers as omega.

Can you understand mathematical symbols? Can you understand d_nn <-->
n?

Writing down a bunch of symbols does not mean that you created a
mathematical notation. Rephrase what precisely "There are more natural
numbers d_nn than natural numbers n." shall mean.

It means that we have discovered an inconsistency in the assumption
that there were infinitely many finite numbers.

Regards, WM

.

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