Re: Cantor Confusion


MoeBlee schrieb:

mueckenh@xxxxxxxxxxxxxxxxx wrote:
MoeBlee schrieb:

David Marcus wrote:
Obviously, I could be wrong, but I think WM means map it on a line of
the list. He seems to think that because we construct the diagonal from
the list, the diagonal must be one of the lines in the list. Why he
thinks this, I have no clue.

You mean map the diagonal (or the range of the diagonal, or whatever)
onto one of the finite sequences that is in the range of the infinite
sequence of those finite sequences? I.e., map the diagonal onto a
member of the range of S? A 1-1 map? If so, yes, I would share your
bafflement as to why we should think there is such a mapping or what
contradiction there is in there not being such a mapping.

There is a mapping of the diagonal on a (each) column.

Okay, if you want to put it that way.

There is no mapping of the diagonal on any line.

Okay.

So there is a difference between the natural numbers (initial segements
of the first column) and the natural numbers (lines). There is not a
bijecton N <--> N?

The diagonal cannot have more elements than the width of the matrix is.

I answered that already. If you define 'the width', then it turns out
to be equal to omega, which is just what the length of the diagonal is.

The diagonal is assumed to exist such that each of its digits exists,
actually. This is established by the mapping on a column. 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,...omega <--> omega+1

If you don't define 'the width', then 'the width' is empty language and
is irrelevent.

The number of elements of the diagonal is assumed to be omega.
That is wrong, because only the supremum is omega.

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.

Regards, WM

.

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