Re: how

Le 13/06/2024 à 15:16, Jim Burns a écrit :

On 6/13/2024 6:55 AM, WM wrote:

Le 12/06/2024 à 23:12, Jim Burns a écrit :

 

If any natural number is undefinable, then
the first undefinable has a definable predecessor.

>
That is your error.

 

The definable numbers are definable and
have definable successors.

 The minimal inductive set contains
all and only finite von Neumann ordinals.

Yes. I call it ℕ_def.

 

You will never get into the dark numbers by counting or defining.

 There is no final finite von Neumann ordinal.

Correct. That is the reason why you cannot leave this collection.

 By 'natural number'  I mean
'finite von Neumann ordinal'.

That implies the existence of a FISON and hence definable number.
Induction means existence of FISONs.

 By ℕ  I mean
minimal inductive set.

That is what I call ℕ_def. By induction we prove that ℵo numbers of ℕ remain before ω.
Regards, WM