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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Fri, 20 Jun 2025 12:00:53 +0300
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On 2025-06-19 14:13:49 +0000, WM said:

> On 19.06.2025 09:16, Mikko wrote:
>> On 2025-06-18 13:48:13 +0000, WM said:
>> 
>>> On 18.06.2025 11:39, Mikko wrote:
>>>> On 2025-06-17 10:57:01 +0000, WM said:
>>>> 
>>>>> On 17.06.2025 12:16, Mikko wrote:
>>>>>> On 2025-06-16 10:50:58 +0000, WM said:
>>>>>> 
>>>>>> The error above is that the expression "the last one" is used without
>>>>>> proving that there is a last one.
>>>>> 
>>>>> If all natnumbers are subtracted, none remains.
>>>> 
>>>> True.
>>>> 
>>>>> If all natnumbers are subtracted in their order, one after the other, 
>>>>> none remains.
>>>> 
>>>> The order does not matter,
>>> 
>>> But the order exists for all defined natural numbers. We know of each 
>>> one its predecessor and its successor, if those are existing.
>> 
>> It is also possible to order them differently. Removing all leaves
>> nothing, whther you remove in some order or another or all at the
>> same time.
> 
> But removing all in their natural or any given order (that does not 
> allow two or more to take the same place) implies a last removed one.

No, it does not. That implication is not acceptable without a proof.

>>>> the result is the same anyway.
>>> 
>>> That means that none remains. I case of known order we know the last 
>>> one subtracted.
>>>> 
>>>> But that does not prove that your "the last one" denotes anything.
>>> 
>>> How can an ordered set be completely subtracted in its given order 
>>> without a last one?
>> 
>> Euclid did not specify how to draw a circle. He merely postulated that
>> for any given centre and radius it can be done.
> 
> The radius is a length that allows to construct every point of the 
> circumference. Further compasses are in use for constructions.

Euclid's postulates say nothing about that.

> 
>> Likewise a set theory
>> does not specify how one set can subtracted from another.
> 
> But it postulates or accepts that an order exists.

Cantor originally did. The ZF, which is the most commonly used formal
set theory, doesn't. The nearest it has is the subset relation, which
is a partial order.

> It merely postulates that it can be done.
> 
> I ask how it can be done. Am I the first? Is it forbidden?

It is outside of the scope of the theory. In a particular interpretation
or application of the theory you can and possibly need to answer that
question.

-- 
Mikko