Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

On 08.10.2024 23:08, Alan Mackenzie wrote:

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

On 07.10.2024 18:11, Alan Mackenzie wrote:

What I should have
written (WM please take note) is:

 

The idea of one countably infinite set being "bigger" than another
countably infinite set is simply nonsense.

 

The idea is supported by the fact that set A as a superset of set B is
bigger than B.

 What do you mean by "bigger" as applied to two infinite sets when one of
them is not a subset of the other?

That is not in every case defined. But here are some rules:
Not all infinite sets can be compared by size, but we can establish some useful rules.
 The rule of subset proves that every proper subset has fewer elements than its superset. So there are more natural numbers than prime numbers, |N| > |P|, and more complex numbers than real numbers, |C| > |R|.  Even finitely many exceptions from the subset-relation are admitted for infinite subsets. Therefore there are more odd numbers than prime numbers |O| > |P|.
 The rule of construction yields the numbers of integers |Z| = 2|N| + 1 and the number of fractions |Q| = 2|N|^2 + 1 (there are fewer rational numbers Q# ). Since all products of rational numbers with an irrational number are irrational, there are many more irrational numbers than rational numbers |X| > |Q#|.
 The rule of symmetry yields precisely the same number of real geometric points  in every interval (n, n+1] and with at most a small error same number of odd numbers and of even numbers in every finite interval and in the whole real line.

The standard definition for infinite (or finite) sets being the same
size is the existence of a 1-1 correspondence between them.
 You seem to be rejecting that definition.  What would you replace it by?
You have specified "bigger" for a special case.  What is your definition
for the general case?
 

Simply nonsense is the claim that there are as many algebraic numbers
as prime numbers.

 It is not nonsense.  The prime numbers can be put into 1-1
correspondence with the algebraic numbers, therefore there are exactly
as many of each.

Nonsense. Only potential infinity is used. Never the main body is applied.

For Cantor's enumeration of all fractions I have given a simple
disproof.

 Your "proofs" tend to be nonsense.

It appears to you because you are unable to understand. Here is the simplest:
Theorem: If every endsegment has infinitely many numbers, then infinitely many numbers are in all endsegments.
Proof: If not, then there would be at least one endsegment with less numbers.
Note: The shrinking endsegments cannot acquire new numbers.
Regards, WM

 

Regards, WM