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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Fri, 6 Jun 2025 12:47:33 +0200
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On 06.06.2025 09:37, Mikko wrote:
> On 2025-06-05 20:36:45 +0000, WM said:
> 
>> On 05.06.2025 09:32, Mikko wrote:
>>> On 2025-06-04 17:32:28 +0000, WM said:

>>> The expression "subtracting them individually" should be represented
>>> mathematically, e.g. a sequence. Informal expressions tend to lead
>>> to bad proofs.
>>
>> This expression is represented by
>>
>> ((((ℕ \ {1}) \ {2}) \ {3}) ...) = { }
> 
> That may be good enough when you want to prove something that we already
> believe anyway. But for a sufficiently rigorous proof of something else
> the "..." should be replaced with something more mathematicsl.

The "..." can be replaced with the singletons of definable natural 
numbers but the result is not the empty set. It is impossible however to 
replace them with the singletons of all natural numbers in order to make 
the result true. Just this is the proof that not all natural numbers are 
definable.

Regards, WM