Re: how

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.