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From: WM <wolfgang.mueckenheim@tha.de>
Newsgroups: sci.logic
Subject: Re: Simple enough for every reader?
Date: Thu, 12 Jun 2025 11:44:06 +0200
Organization: A noiseless patient Spider
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In-Reply-To: <102e1f6$2ilof$1@dont-email.me>

On 12.06.2025 10:00, Mikko wrote:
> On 2025-06-11 11:38:52 +0000, WM said:
> 

>> Outside of ZF in correct mathematics this proof for all definable 
>> numbers n
>> |ℕ \ {1}| = ℵo.
>> |ℕ \ {m ∈ ℕ | m < n}| = ℵo
>> ==> |ℕ \ {m ∈ ℕ | m < n+1}| = ℵo.
>> shows that is impossible to extend definability to all natural numbers 
>> with none remaining undefined.
> 
> It does not show that. The "proof" does not even mention definability.

"this proof for all definable numbers n"

> The conclusion follows from the second sentence alone when n is understood
> to be universally quantified, so the first sentence is not needed and 
> should
> not be there.

It is there. Only definablenumbers can be quantified as individuals.

Regards, WM
>