Re: Cantor Confusion

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

1
11
111
...

The length of each column is omega.

I assume that we are using ZFC as our logical system.

Assuming by the "length of each column" that you mean the number of 1's
in each column, then the number of 1's in each column has cardinality
aleph_0. So, what you wrote is essentially correct.

The length of each line is less than omega.

Each line has a finite number of 1's. Finite cardinality is less than
aleph_0. So, what you wrote is essentially correct.

The length o the diagonal is less than omega.

The number of 1's in the diagonal is aleph_0. So, what you wrote is
false. In fact, I have no clue what you could possibly be thinking that
would lead you to make such an obviously incorrect statement. The length
of the diagonal is clearly the same as the length of the first column,
and you just wrote above that the length of each column is omega.

The length of the diagonal is clearly not more than the length of any
line.

Do you disagree that the length of the diagonal is the same as the
length of the first column?

No it is the same. I distinguish beween lines and columns.

OK. The length of the diagonal is the same as the length of the first
column.

As for your statement that "the length of the diagonal is clearly not
more than the length of any line", the length of the first line is 1,
and this is less than the length of the diagonal, so your statement as
written is false.

Maybe that my use of "any" was wrong. Please replace it by "every": the
length of the diagonal is clearly not more than the length of every
line.

So, you are saying that there is some line whose length is greater than
or equal to the length of the diagonal. Is that correct?

Since the length of the diagonal equals the length of the first column,
you are also saying that there is some line whose length is greater than
or equal to the length of the first column. Is that correct?

--
David Marcus
.

Relevant Pages

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  • Re: Cantor Confusion
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  • Re: Two results of set geometry
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    (sci.math)