Re: how

On 6/12/2024 4:33 PM, WM wrote:

Le 12/06/2024 à 20:54, Jim Burns a écrit :

On 6/12/2024 2:27 PM, WM wrote:

Le 12/06/2024 à 20:18, Jim Burns a écrit :

On 6/11/2024 10:44 AM, WM wrote:

ℕ \ {1, 2, 3, ...} = ?
Where are  the followers?

>
ℕ\{0,1,2,…}  =  ∅

>
So there are no followers?

>
there is no j ∈ ℕ after ℕ

>
i.e., after all natural numbers.

There is no natural number after (≥)
all natural numbers.
ℕ = {y: ∀₃X⤾⁺¹₀:X∋y}
ℕ is the minimal inductive meta.set.
∀ᴺx ∃ᴺy≠x: x<y  ⇔
¬∃ᴺy ∀ᴺx≠y: x<y

That means:
If every number is subtracted,
then no successors remain.

If every number is deleted,
then every number is deleted.

If only definable numbersa are subtracted, then successors remain.

Only if some natural number is undefinable.
If any natural number is undefinable, then
the first undefinable has a definable predecessor.
No undefinable has a definable predecessor.
No natural number is undefinable.
After all definables are deleted from ℕ
no successors (no anything) remain in ℕ

ℕ is the minimal inductive meta.set.

>
ℕ is all natural numbers.
Not more and not less.

Proposal 1.
Definitions are only
statements of _what the definer means_
Without evidence to the contrary,
the definer is presumed to be
honest and aware of what they mean,  and
definitions are presumed to be
true statements of what they mean.
On questions beyond what the definer means,
their definitions do not receive
a presumption of truth,
but they remain free to argue their POV.
By 'ℕ'  I mean the minimal inductive meta.set.