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

On 04.04.2025 18:41, Alan Mackenzie wrote:

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

On 03.04.2025 21:10, Alan Mackenzie wrote:

 

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.

 

No.

 You're wrong.

You are caught in a world of stupidity. Set theorists have damaged the honour of human intellect even more than Pope Pius XII.
When an element is added to a set, then this set is no longer the same but different because the number of its members is different.

This is easily contradicted by observing that 1/2 is not a natural
number while all natural numbers are fractions.

 It is not contradicted.  There is a 1-1 correspondence between positive
fractions and natural numbers.

Only for natural numbers which have less predecessors than successors.

I understand that you are duped. And I have explained why. Every pair of
the bijection has almost all elements as successors.

 Eh??  What the heck are you going on about?  Hint: a bijection is a set
of pairs.  It is not ordered.

The natural numbers are ordered. Therefore the bijection is ordered.
"thus we get the epitome (ω) of all real algebraic numbers [...] and with respect to this order" Georg Cantor – Gesammelte Abhandlungen mathematischen und philosophischen Inhalts", Springer, Berlin (1932) p. 116]

Every natural number that you can use in a bijection has finitely many
predecessors but infinitely many successors which will never be used.

 Wrong.

Name an exception.

Just this is excluded. Only definite positions are admitted. No evasion
into the infinite.

 You mean, you're only allowing a finite number of transpositions?

No, but all the infinitely many indices of transpositions are finite.
Regards, WM
   In

that case, the distinguished element indeed does not disappear.
 

Regards, WM