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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Sat, 31 May 2025 15:47:51 +0200
Organization: A noiseless patient Spider
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On 31.05.2025 12:11, Mikko wrote:
> On 2025-05-30 14:46:55 +0000, WM said:
> 

>>>> {1, 2, 3, ..., n, n+1} has infinitely many (ℵo) successors is a 
>>>> consequence of {1, 2, 3, ..., n} has infinitely many (ℵo) 
>>>> successors, and ℵo - 1 = ℵo.
>>>
>>> Still no proof.
>>
>> Proof that every definable natural number has more successors than 
>> predecessors.
> 
> You have not shown that proof, either.

I have above. You cannot understand it. That is a different thing.

> But because every natural number
> has more successors than predessors it hardly is interesting, unless
> you happen to have an interesting way to say 'more'.

Every definable natural number has finitely many predecessors but 
infinitely may successors. Since all successors can be reduced to the 
empty set by subtractig them collectively,
ℕ \ {1, 2, 3, ...} = { }	
they could also be reduced to the empty set by subtracting them 
individually - if this was possible. But then the well-order would force 
the existence of a last one. Contradiction.

Regards, WM