Re: The non-existence of "dark numbers"

Ben Bacarisse <ben@bsb.me.uk> wrote:

Alan Mackenzie <acm@muc.de> writes:

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

On 12.03.2025 18:42, Alan Mackenzie wrote:

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

If the numbers are definable.

Meaningless.  Or are you admitting that your "dark numbers" aren't
natural numbers after all?

They

They?

Learn what potential infinity is.

I know what it is.  It's an outmoded notion of infinity, popular in the
1880s, but which is entirely unneeded in modern mathematics.

That makes "modern mathematics" worthless.

What do you know about modern mathematics?  You may recall me challenging
others in another recent thread to cite some mathematical result where
the notion of potential/actual infinity made a difference.  There came no
coherent reply (just one from Ross Finlayson I couldn't make head nor
tail of).  Potential infinity isn't helpful and isn't needed anymore.

WMaths does (apparently) have one result that is not a theorem of modern
mathematics.  In WMaths there sets P and E such that

  E in P   and   P \ {E} = P

WM himself called this a "surprise" but unfortunately he has never been
able to offer a proof.

Surprise indeed, but no surprise, too.  He's coming pretty close to
assertions like that in his replies to me, in the bits which are
coherent.

On another occasion he and I came close to another when I defined a
sequence of rationals that he agreed was monotonic, increasing and
bounded above but which (apparently) does not converge to a real in
WMaths.  It was "defined enough" to be monotonic and bounded but not
"defined enough" to converge to a real.

Maybe it converged to some function of dark numbers.  They're not real,
are they?  ;-)  But he's clearly ignorant of the axiom of completion (the
one which distinguishes the reals from ordered subfields).

But, in general, he hates talking about his WMaths because it gets him
into these sorts of blind alleys.

But he loves going on about "dark numbers" and apparently still believes
they exist.  Hmmm.

--
Ben.

--
Alan Mackenzie (Nuremberg, Germany).