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

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?

Thus in Hilbert's hotel, although all the rooms are occupied, one of
these rooms can be vacated to make room for a new guest without
expelling an existing guest.

Real fools are really delighted by counterintuitive results.

You've run out of mathematics, so you resort to ad hominem.  Not very
good.  Hint: the person who devised "Hilbert's hotel" was in no sense a
fool.

Adding that new guest doesn't change the number of guests in the
hotel, or the number of rooms required.

Real fools are really delighted by counterintuitive results.

See many of Jim's posts over the last few days for details.

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.

Regards, WM

--
Alan Mackenzie (Nuremberg, Germany).