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 28.03.2025 16:45, Alan Mackenzie wrote:

WM <invalid@no.org> wrote:

Am 26.03.2025 um 22:38 schrieb Alan Mackenzie:

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

On 26.03.2025 21:06, joes wrote:

Am Wed, 26 Mar 2025 20:36:40 +0100 schrieb WM:

The potential infinite is a variable finite. Cantor's actual
infinity is not variable but fixed. (Therefore Hilbert's hotel
is potential infinity.)

What we refer to as infinite isn't variable.

The number of guests/rooms in Hilbert's hotel is infinite but can
grow.  That is variable infinity.

You are mistaken.  (Countable) infinity stays the same when you add
finite and countably infinite numbers to it.

That proves that cardinality is nonsense. When a new guest arrives,
then the number of guests grows by 1.

Yes indeed.  There were aleph-0 guests beforehand, the new guest arrives
growing that number by 1, giving aleph-0.  Why do you find this so
difficult to understand?

It is not at all difficult to understand. Difficult to understand is
only why cardinality is used at all.

Those two sentences contradict eachother.  Cardinality is used because it
is a sensible way of comparing the size of sets.

It is worthless because it cannot describe changes of substance. If
there are |ℕ| natural numbers, then there are |ℕ|^2 positive fractions.

Yes, and aleph_0^2 = aleph_0.  There are as many positive fractions as
natural numbers.  This was proven by Cantor.  That you don't understand
the proof is your problem, not ours.

The cardinality is the same because it counts only the first elements.

That's a meaningless concatenation of words.  Cardinalities do not count.
What you might mean by "only the first elements" is completely obscure.

Potential infinity. Otherwise it could not overlook the big difference.

Just "infinite" is a lot simpler, hence is used in modern mathematics.

There is only one important detail, namely that lossless exchanges
cause losses. It is sufficient to reject every intelligent being.

An infinite process of lossless exchanges can cause loss, as we have
seen.

No. You have not seen it.

What makes you think you know what I have and have not seen?

You are mistaken and try to maintain your mistakes by "limits" which
are not used in Cantor's theory:

I'm quite sure Cantor was enirely familiar with the theory of limits.  It
seems you are not.

"so daß jedes Element der Menge an einer bestimmten Stelle dieser Reihe
steht" [E. Zermelo: "Georg Cantor – Gesammelte Abhandlungen
mathematischen und philosophischen Inhalts", Springer, Berlin (1932) S.
152]

"So that each element of the set stands at a definite position of this
sequence."  That has no relevance to anything at issue here.  In
particular, it has no relevance to the loss of your favourite set element
caused by an infinite sequence of transpositions.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).