Re: Incompleteness of Cantor's enumeration of the rational numbers (extra-ordinary)

WM <wolfgang.mueckenheim@tha.de> wrote:

On 05.01.2025 12:28, Alan Mackenzie wrote:

WM <wolfgang.mueckenheim@tha.de> wrote:

[ .... ]

Finally, the most familiar example is this: The (magnitudes of)
natural numbers are potentially infinite because, although there is
no upper bound, there is no infinite (magnitude of a) natural number.

There are no "actual" and "potential" infinity in mathematics.

It has been exorcized by those matheologians who were afraid of the
problems introduced to matheology by these precise definitions.

Not entirely sure what you mean by matheologian.  As I say below ...

The notions are fully unneeded, and add nothing to any mathematical
proof.  There is finite and infinite, and that's it.

When I did my maths degree, several decades ago, "potential infinity"
and "actual infinity" didn't get a look in.  They weren't mentioned a
single time.

That has opened the abyss of nonsense to engulf mathematics with such
silly results as: A union of FISONs which stay below a certain
threshold can surpass that threshold.

You're mistaken here and likely lying.  The only person talking about
FISONs "below a certain threshold" is you.  Everybody else here is
talking about the union of ALL FISONs.  Correct is that the union of all
FISONs is N.  You seem unable to understand this.

The only people who talk about "potential" and "actual" infinity are
non-mathematicians who lack understanding, and pioneer mathematicians
early on in the development of set theory who were still grasping
after precise notions.

All mathematicians whom you have disqualified above are genuine
mathematicians.

Maybe, but they weren't discussing modern mathematics in the excerpts
you've quoted.  At least one of them confused the size of a set with the
size of its elements, which suggests strongly (s)he is not a
mathematician.

What you

What you too!

--
Alan Mackenzie (Nuremberg, Germany).