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

Ross Finlayson <ross.a.finlayson@gmail.com> wrote:

On 01/07/2025 03:36 AM, Alan Mackenzie wrote:

Moebius <invalid@example.invalid> wrote:

Am 05.01.2025 um 12:28 schrieb Alan Mackenzie:

[ .... ]

None of the above extracts is about mathematics.  If there were such
things as "potential" and "actual" infinity in maths, then they would
make a difference to some mathematical result.  There would be some
theorem provable given the existence of PI and AI which would be false
or unprovable with just plain infinite, or vice versa.  Or something
like that.  Nobody in this discussion has so far attempted to cite
such a result.

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.  Instead, precise definitions were given to
"finite" and "infinite", and we learnt how to use these definitions
and what could be done with them.

Sure. But this needs a context; usually (some sort of) set theory.

The only people who talk about "potential" and "actual" infinity are
non-mathematicians who lack understanding

Like those mentioned above?

OK, and mathematicians in their time off.  ;-)

Oh, how about yin-yang ad-infinitum,
a simple and graphical example of
something that is, in the infinite,
not what it is, in the limit,
that it's actual infinite limit,
differs its potential infinite limit.

What do you mean by the "yin-yang ad-infinitum", exactly?

I'm not even sure how you would define, or distinguish "actual infinite
limit" and "potential infinite limit".  Do such things already have
accepted mathematical definitions?

It's also well known that a limit frequently has properties not in any of
the terms.  For example, the limit of a sequence of rational numbers is
often irrational.  Is this what you meant?

[ .... ]

--
Alan Mackenzie (Nuremberg, Germany).