Re: The non-existence of "dark numbers"

WM <invalid@no.org> wrote:

Am 26.03.2025 um 22:34 schrieb Alan Mackenzie:

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

On 26.03.2025 07:24, Jim Burns wrote:

"WM-logic" says that lossless exchanges at finite steps are lossless
....

Nobody has contradicted this, that I'm aware of.  It's the accumulation
of _all_ of these lossless exchanges where unexpected things happen.

If they happen, then there is a first instance where they happen. Every
n.e. subset of a countable set has a first element.

The set of integer steps at which a loss occurs is empty.  It thus has
no least member.  It is only in the infinite limit where the loss
occurs.

If I recall correctly, this "lossless exchange" scenario concerns a
sequence of swaps, where every few steps some distinguished item moves
from its current place n to its new place n + 1.

Try using your intuition correctly, for once.  This item is steadily
moving to higher numbered places, without limit.  For any place, the
item will sooner or later pass that place.  In the limit, it passes
_all_ places.  In informal language, it "disappears off to infinity",
and thus is no longer at one of the numbered places.

.... and is the foundation of rational thinking.

I think rational thinking has more general foundations than
pontifications about lossless exchanges.

But that is one necessary foundation. When it breaks down, then thinking
breaks down.

You base your mathematical thinking on faulty intuition.  You do not
base it on the axioms and logic which have chrystallised out of a lot of
very clever thinking over the last few centuries.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).