Re: how

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.