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

On 05.01.2025 12:28, Alan Mackenzie wrote:

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

The set of these prime gaps is infinite, without qualification.  Euclid
could have told you that.

Euclid did not believe in actual infinity. The prime gaps have no upper limit.

 

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.

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.

 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 genuin mathematicians.
What you