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 22.03.2025 16:28, Alan Mackenzie wrote:

The term "potentially infinite" has no use in
mathematics.  Philosophers, etc., might cling to it, though.

For your education.:

"On account of the matter I would like to add that in conventional
mathematics, in particular in differential- and integral calculus, you
can gain little or no information about the transfinite because here the
potential infinite plays the important role, I don't say the only role
but the role emerging to surface (which most mathematicians are readily
satisfied with). Even Leibniz with whom I don't harmonize in many other
respects too, has [...] fallen into most spectacular contradictions with
respect to the actual infinite." [G. Cantor, letter to A. Schmid (26 Mar
1887)]

[ snip seven more of the same ]

Will you show gratitude to be educated in great detail?

What you have cited in no way contradicts what I said above.  I repeat,
"potentially infinite" has no use in mathematics.  You did not, in your
voluminous post, cite any indication of a _use_ of "potentially
infinite", only some philosophising about it.

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".

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

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).