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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Thu, 26 Jun 2025 13:36:14 +0300
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On 2025-06-25 20:08:06 +0000, WM said:

> On 25.06.2025 09:53, Mikko wrote:
>> On 2025-06-24 17:33:20 +0000, WM said:
>> 
>>> On 24.06.2025 11:15, Mikko wrote:
>>>> On 2025-06-23 10:59:08 +0000, WM said:
>>> 
>>>>> The natural numbers are well-ordered, from the first to the last which 
>>>>> is subtracted. Does this change if all can be subtracted?
>>>> 
>>>> There is no last natural number. Every subset of natural numbers
>>>> has a first member (in the arithmetic order) but infinite subsets
>>>> don't have a last one.
>>> 
>>> Then it is impossible to remove all of them in their order.
>> 
>> Maybe so. There expression "to remove all of them in their order" is
>> an informal expression that might refer to something that can be
>> expressed mathematically but maybe it does not.
> 
> You confuse mathematics with formalism. The greatest mathematicians 
> have lived before formalism had been invented. Formalism is only a 
> crutch for those who cannot think without crutches.

Formalism was invented to avoid or correct the errrors the greatest
mathematicians made before formalism was invented. The invoention
of formalism was not a sudden event but a gradual process that took
thousands of years.

> If all naturals can be subtracted then this can be done in their order. 
> For enumerating another set the order is even a precondition.

For enumeration an order is necessary but not sufficient. But for
subtraction it is not. For example, irrational numbers have their
arithmetic order but cannot be enumerated. But they can be sutracted
from the set of real numbers and the result is the set of rational
numbers.

-- 
Mikko