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

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.

 

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

This was proven by Cantor.  That you don't understand
the proof is your problem, not ours.

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

 

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

 That's a meaningless concatenation of words.

It is a pity that you can't understand. Every natural number that you can use in a bijection has finitely many predecessors but infinitely many successors which will never be used.

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?

I know your mistakes and their origin.

 

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.

But he did not use them in his bijections.

 

"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.

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