Re: Cantor Confusion


William Hughes schrieb:

mueck...@xxxxxxxxxxxxxxxxx wrote:
William Hughes schrieb:

Each diagonal digit is formed by a line.

and by a column!

Yes, and the number of columns is equal to the number of lines.
The fact that no single line contains all the columns is not
important.


The fact that *no* set of lines contains all the line is not important
too.

I think we have obtained agreement, that we disagree in this point and
that further discussion will not lead to a result acceptable for both
parties. The final state is:

1) The number of columns is equal to the number of lines.

2) Both are infinite, according to set theory this means omega or
aleph_0.

But you believe: omega or aleph_0 is the maximum of the set of lines
and the supremum of the set of columns, which, however, is not taken by
any column. You believe the different meaning is not important.

I believe that a taken maximum and a not taken supremum are so
different (although the difference is very tiny) that set theory,
necessarily assuming this, is wrong.

The difference between maximum und supremum is, by the way, the reason
why I insist that an actually infinite set of natural numbers must
contain a non-natural number.




D can be projected or mapped into a line. Then it is a line longer than
any line.


D cannot be projected or mapped into any line.

You can map d_kk --> d_k.


You wish to introduce a line with omega letters?
That yields what I always say: There is no infinite number of natural
numbers without an infinite number omega.

All you are doing is calling the infinite number of natural numbers
omega.
Then you say, there is no omega without omega.

I think this is obvious.

You do not want to
merely assume
that omega doesn't exist. You want to show that assuming its existence
leads
to a contradiction.

If we do not intermingle supremum and maximum, it does so in fact.

You don't get a contradiction if you assume that a supremum is taken
which, by definition, is not taken?


What definition says that a supremum is not taken?

The definition of natural numbers as finite numbers: A n : n < omega.

Regards, WM

.

Relevant Pages

  • Re: Cantor Confusion
    ... omega or aleph_0 is the maximum of the set of lines ... > and the supremum of the set of columns, which, however, is not taken by ... > why I insist that an actually infinite set of natural numbers must ... Again this is true but there is no contradiction. ...
    (sci.math)
  • Re: Two results of set geometry
    ... If one assumes that this supremum does not exist, ... It shows that there is no infinite height. ... No the set of heights is 1,2,3,.. ... It contains omega number and omega finite intial segments. ...
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  • Re: Cantor Confusion
    ... This supremum does not have to be the ... Then omega is the *supremum* ... The set of lines has infinitely many elements, ... Why would William disagree that there is a set of lines? ...
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  • Re: Cantor Confusion
    ... This supremum does not have to be the ... Then omega is the *supremum* ... The set of lines has infinitely many elements, ... then we agree that actual infinty does not exist. ...
    (sci.math)
  • Re: Cantor Confusion
    ... Why is the supremum "not sufficient" to define the length of the ... (Note, there is no line with an infinite index, so ... your answer should not depend on a line with an infinite index)? ... Every column has a fixed number omega of terms. ...
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