Re: The reality of sets, on a scale of 1 to 10 [Was: The non-existence of "dark numbers"]

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

On 24.03.2025 21:12, Alan Mackenzie wrote:

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

On 23.03.2025 20:39, Alan Mackenzie wrote:

You did not, in your
voluminous post, cite any indication of a _use_ of "potentially
infinite", only some philosophising about it.

Have you not read Hilbert and Cantor? In analysis potential infinity and
only it is used.

What is used is the infinite.  It needs no redundant qualifier
"potential".

You don't grasp it.

I do, very much so.  You don't grasp its redundancy.  I'm trying to help
you understand it.

You have dishonestly snipped the core of my last post.  Here it is
again:

In modern mathematics there are the notions finite and infinite.
They are useful.  I challenge you to produce a theorem which cannot
be proven with those notions, yet can be proven with, additionally,
"potentially infinite".

I told you that all mathematics is based on potential infinity.

You were wrong.  Unless you can produce that theorem which requires the
notion of "potentially infinite".

If neither you nor anybody else can do this, then we must conclude
that "potentially infinite" has no use in mathematics.

Actual infinity has no use in any kind of applied mathematics.

Neither "potential" nor "actual" are necessary qualifications of
infinite.  They just cause confusion.  The term infinite stands on its
own.

In fact all meaningful and correct applications of infinity in
mathematics concern potential infinity, because actual infinity
either is a chimera only or it is dark and therefore cannot be
manipulated and applied in mathematics.

"Potential infinity", I repeat, is unnecessary,

Try to understand the scholars I quoted.

None of these scholars in your cites were doing mathematics with
"potential infinity".  They were just philosophising.  Maybe "potential
infinity" has a role to play in philosophy.  I wouldn't know.  It has no
role in mathematics.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).