Re: Cantor Confusion


MoeBlee schrieb:



As I said, I don't care about (3).

My mathematical notions include the fact that the diagonal of a matrix
cannot have more elements than every line. If in your opinion this
position is not valid in ZFC or in your logic, then we should stop
here.

The diagonal is assumed to exist such that each of its digits exists,
actually. This is established by the mapping on a column.

No it is not.

But it cannot
be established by the mapping on any line. You should recognize that
the following bijection between columns and lines shows a
contradiction, because one element is missing:

1 <--> 1
1,2 <--> 2
1,2,3 <--> 3
...
1,2,3,...n <--> n
...
1,2,3,... <--> omega

1, 2, 3 ... is NOT a line.

But it is the whole column which would correspond to the line omega
(which does not exist).

And it is NOT a contradiction that 1, 2, 3, .. omega is equinumerous
with omega+1.

Of course, the set 1,2,3... is equinumerous to the set 2,3,4,..., 1.
Nevertheless the diagonal of a matrix cannot have more elements than
every line. If in your opinion this position is not valid in ZFC or in
your logic, then we should stop here.

I answered that already. And the cardinality of the diagonal is not
assumed, but is proven, to be omega.

It is *assumed* by stating the axiom of infinity. Without this
assumption the length was not omega.

It is PROVEN from the axiom of infinity.

But the existence of this axiom is assumed.

since I would not have bothered to even post
a proof about denumerable sequences and talk with you about it for so
long if I accepted any condition that I can't use the axiom of
infinity.

That is *not my condition* but it is the result which follows from the
fact that the diagonal cannot be longer than every line. If I have the
choice either to accept the axiom of infinity with the condition that a
diagonal can be longer than every line, or to drop both notions, then I
choose the second. If you can live with the contrary, then try to do
it. I wish you nice dreams.

Regards, WM

.

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