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

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

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.
 

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.

No, that is a subset relation („reality”). The identity function is
obviously not a bijection from N to Q.

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.

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.

"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.

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.