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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Fri, 27 Jun 2025 21:36:41 +0200
Organization: A noiseless patient Spider
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On 27.06.2025 09:33, Mikko wrote:
> On 2025-06-26 13:09:32 +0000, WM said:

>>
>> If we subtract in the order that is used for enumerating then a last 
>> one is necessary.
> 
> No, there is no last one in an infinete enumeration.

Then it is not finished or completed.

> The word "infinite"
> originally meant "having no end".

That is true. But if infinite sets are complete, then they are infinite 
only because the end is not visible.

> We can remove all odd numbers from the
> natural numbers, leaving the even numbers, but there is no last number
> removed.

Then not all are removed. All completely, never ending and in order 
implies a contradiction.
> 
>>> For example, irrational numbers have their
>>> arithmetic order but cannot be enumerated.
>>
>> The reason is that infinite sets cannot be enumerated. Also there 
>> completion would necessitate a last one.
> 
> Infinite enumerable sets can. Then the enumeration is an infinte sequence.

Then it is never completed. Going on and on is not a proof of completeness.
> 
>> < But they can be sutracted
>>> from the set of real numbers and the result is the set of rational
>>> numbers.
> 
>> That can be done collectively only.
> 
> Doesn't matter.
> 
That is very important because these things are confused very often.

Regards, WM