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

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. It is worthless because it cannot describe changes of substance. If there are |ℕ| natural numbers, then there are |ℕ|^2 positive fractions. The cardinality is the same because it counts only the first elements. Potential infinity. Otherwise it could not overlook the big difference.

 

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