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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