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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.math
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (infinitary)
Date: Tue, 8 Oct 2024 17:21:16 -0400
Organization: i2pn2 (i2pn.org)
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On 10/8/24 3:51 PM, Chris M. Thomasson wrote:
> 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...
> 

Nor, as he tries to do, distinguish between the sizes of those sets. 
WHen you take a finite number of elements from a finite set, it gets 
smaller.

If you take a finite number of sets, or even some infinte subsets, you 
can still be left with an infinite set of exactly the same size.

> 
> 
> 
>>
>> 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.
>