Re: Cantor Confusion


Franziska Neugebauer schrieb:

mueckenh@xxxxxxxxxxxxxxxxx wrote:

William Hughes schrieb:
Let he original matrix be A.
[...]
1
12
123
...

Something is missing here:

1uuu...
12uu...
123u...
...

No. We discuss the question whether an infinite set of naturl numbers
requires an infinite number as an element. The naural numbers are
expressed by the digits of a line. We could also use

1
11
111
....

The "u" are not appropriate.

Usually a matrix m(a,b) is a function of domain A x B (it is
rectangular not a triangle) and some co-domain. Here A equals B equals
omega by definition. A matrix with domain A x B is called (generalized)
square iff A = B.

"u" stands for undefined (empty), not set. The "triangle" you see is not
the structure of the matrix but its occupancy.

We discuss just this triangle as our marix.

Prima facie "Adding x to each column" ("at the end") is not "possible",
since the columns have no end (no last element).

If 1,2,3,... has the ordinal number omega and if it is possible to
construct 2,3,4,...,1 and if it is meaningful to denote the ordinal
number of 2,3,4,...,1 by omega + 1, then your argument fails. But just
these antecedents are assumed.

Regards, WM

.

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