Re: Cantor Confusion


William Hughes schrieb:

mueckenh@xxxxxxxxxxxxxxxxx wrote:
David Marcus schrieb:

mueckenh@xxxxxxxxxxxxxxxxx wrote:
David Marcus schrieb:
mueckenh@xxxxxxxxxxxxxxxxx wrote:
David Marcus schrieb:

The set of natural numbers is an infinite set that contains only finite
numbers.

Please do not assert over and over again this unsubstantiated nonsense
(this word means exactly what you think) but give an example, please,
of a natural number which does not belong to a finite sequence. If you
cannot do so, then it is obviously unnecessary to consider N as an
infinite sequence, because all its members belong to finite sequences.

I didn't say anything about sequences, finite or otherwise. So, your
request is irrelevant to my statement.

The sequence of natural numbers is not comprehensible in ZFC? Neither
is the sequence of partial sums of a converging series? Nor are the
finite sequences which are called (initial) segments of sequences which
are ordered sets. Also the expression "extended sequence" for an
uncountable ordered set is new to you?

Non sequitor.

?
I did not yet conclude anything but asked some questions.

Let's make it simple. I'll give a statement and you say whether you
think it is provable in ZFC. Is

The set of natural numbers is infinite

provable in ZFC? Please answer "Yes" or "No".

Don't you think that you should label all your posts as
"NON-STANDARD MATHEMATICS"?

Cantor invented omega and defined omega as a whole number.
Who changed this standard meaning?
Why do you think this meaning was changed?
When do you think the contrary meaning became standard?
What is the contrary meaning?
Do you agree that A n: n < omega is incorrect?
If not, why do you complain about non-standard meaning on Cantor's
definition of omega as a whole number?

Since Cantor predates axiomatic set theory, if you write anything that
uses Cantor's definitions without checking whether the definitions are
still standard is "Non-Standard Mathematics".

Therefore I put above list of questions in order to find out what your
understanding of the standard is. If you say: My position is standard,
that is fine for you, but it is not sufficient to show anything but
orthodoxy.

If you want to discuss
history, that is fine, but you should label your posts as such. This is
simple courtesy. If you use words without defining them, readers assume
you are using them in their current meanings. If you are using
historical meanings, then either say so or use a different word.

As for the current definition of omega, Kunen's book is a good
reference. According to Kunen, omega is not a natural number.

That is out of any question.

I'm
guessing that by "whole number" you mean natural number, but I really
don't know, since you seem to have your own language for everything and
you never give definitions for any of the words that you use.

"Whole number" is Cantor's name for his creation.

My question is : Do you maintain omega > n for all n e N? I know that
modern set theory says so. If something can be larger than a number,
then it must be a number.

Omega is not an element of N. However you can compare omega with
any element of n.

Therefore we can compare the diagonal with every line. We find that the
diagonal is longer than every line. This is about the same as the vase
which is empty at noon or Tristram Shandy who completes his diary.

No reason to bother about this.

Regards, WM

.

Relevant Pages

  • Re: Cantor Confusion
    ... infinite sequence, because all its members belong to finite sequences. ... Cantor invented omega and defined omega as a whole number. ... Why do you think this meaning was changed? ... When do you think the contrary meaning became standard? ...
    (sci.math)
  • Re: Cantor Confusion
    ... infinite sequence, because all its members belong to finite sequences. ... Cantor invented omega and defined omega as a whole number. ... Why do you think this meaning was changed? ... When do you think the contrary meaning became standard? ...
    (sci.math)
  • Re: Cantor Confusion
    ... infinite sequence, because all its members belong to finite sequences. ... Cantor invented omega and defined omega as a whole number. ... Why do you think this meaning was changed? ... When do you think the contrary meaning became standard? ...
    (sci.math)
  • Re: Cantor Confusion
    ... of a natural number which does not belong to a finite sequence. ... Cantor invented omega and defined omega as a whole number. ... Who changed this standard meaning? ... In the sequence of ordinals, starting with and minimally closed under ...
    (sci.math)
  • Re: Cantor Confusion
    ... infinite sequence, because all its members belong to finite sequences. ... Cantor invented omega and defined omega as a whole number. ... Why do you think this meaning was changed? ... When do you think the contrary meaning became standard? ...
    (sci.math)
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