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From: "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Tue, 8 Oct 2024 12:51:54 -0700
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On 10/8/2024 5:29 AM, Richard Damon wrote:
> On 10/8/24 6:22 AM, WM 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. Simply nonsense is the claim that there are as many 
>> algebraic numbers as prime numbers. For Cantor's enumeration of all 
>> fractions I have given a simple disproof.
>>
>> Regards, WM
>>
> 
> Which just shows that you logic system is based on contradictory 
> definitions and has blown itself up in them, because you have a bad 
> definition for "size" of an infinite set.
> 
> By your logic, a set can be bigger than a set with just the same number 
> of elements.  BOOOM
> 
> Simple example, look at the set of the integers {1, 2, 3, 4, ...} and 
> the set of the even numbers {2, 4, 6, 8, ...}
> 
> By your logic the even set must be smaller, but we can transform the set 
> of integers to a set of the same size by replacing every element with 
> twice itself (an operation that can't affect its size) and we get that 
> same set of even numbers.

In the sense of, there are infinitely many evens and odds... Just like 
there are infinitely many natural numbers. The infinite "pool" to draw 
from, in a sense. Classifying a number (even odd) from an infinite pool 
of them does not magically make the infinite suddenly become finite by 
any means...




> 
> Thus the set of even numbers (obtained by doubling the integers) is the 
> same set as the set of even numbers (obtained by removing the odd 
> integers) but is bigger than it.
> 
> Yes, there are other forms of mathematics where the set of even numbers 
> is smaller than the set of the integers, but in those mathematics you 
> can't do other things, like replace a set with another set of exactly 
> the same size by changing all the values, but that isn't your 
> mathematics either, yours has just gone BOOM because it got "too full" 
> with an infinite set.