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From: FromTheRafters <FTR@nomail.afraid.org>
Newsgroups: sci.math
Subject: Re: how
Date: Sat, 08 Jun 2024 20:40:39 -0400
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Jim Burns laid this down on his screen :
> On 6/8/2024 8:42 AM, WM wrote:
>> Le 07/06/2024 à 23:00, Jim Burns a écrit :
>
>>>>> I give a description of an individual number j in ℕ⁺
>>>>> -- a description which does not distinguish between
>>>>> different numbers in ℕ⁺ 
>>>>> 
>>>>> ⎛ Each number in ℕ⁺ has a successor.
>>>>> ⎜ Each nonzero number in ℕ⁺ has a predecessor.
>>>>> ⎝ Each nonempty subset of ℕ⁺ holds a first number.
>
>>> if ℕ⁺\Defble is not.empty
>>> then
>>> ℕ⁺\Defble holds first j ∈ ℕ⁺
>>> j ∉ Defble
>>> j-1 ∈ Defble
>>
>> That is your error.
>
> No.
> That is correct about _what I have described_
>
>> If j ∈ Defble then j^j^j ∈ Defble.
>> Nevertheless j^j^j^j^j is finite, but there are ℵo undefinable natural 
>> numbers.
>
> No.
> There are 0 first undefinable natural numbers.
> There are 0 undefinable natural numbers.
>
>> This could only be disproved by defining them.
>
> No.
> It has been disproved by
> giving a description of an individual number j in ℕ⁺
> -- a description which does not distinguish between
> different numbers in ℕ⁺
> and then augmenting the description with
> only not.first.false claims.
>
> None of those claims is first.false for
>   any individual number in ℕ⁺
> None of those claims is false for
>   any individual number in ℕ⁺
>
>> But they will never be defined
>
> Irrelevant.
> Numbers between 0 and Avogadroᴬᵛᵒᵍᵃᵈʳᵒ
> will never be all defined.
> Numbers between 0 and Avogadroᴬᵛᵒᵍᵃᵈʳᵒ
> are all definable.
>
> An apple can be
>   edible and not eaten.
> A tree falling in the forest can be
>   audible and not heard.
> A natural number can be
>   definable and not defined.

But can you have a non-empty set of undefined natural numbers? 
Distinguishable but not distinguished? Discernible yet not discerned? 
One should be able to discern each and every element at least enough to 
decide on membership. Ghost naturals might exist somewhere, but are not 
a subset of the naturals. WM seems to just 'not get' real numbers.