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

On 13.04.2025 10:25, joes wrote:

Am Thu, 03 Apr 2025 22:28:44 +0200 schrieb WM:

On 03.04.2025 21:10, Alan Mackenzie wrote:

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.

No,

Yes.

 

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.

Bijections aren’t ordered.

Bijections with ℕ are ordered by the well-ordered set ℕ.

 

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.

No, a bijection can be an infinite set of pairs - must be, if the sets
are infinite.

Every pair of the bijection has almost all elements as successors.

 

"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.

Then you cannot talk about the limit.

So it is. Bijections with ℕ are well-ordered and have no limit but only all definable terms.
Regards, WM