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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Wed, 9 Oct 2024 14:48:17 +0200
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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
>