Re: how

On 6/13/2024 10:45 AM, WM wrote:

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

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.

We don't need to leave the collection.
We have a telescope which
we use to explore where we cannot go.
The telescope is a finite claim.sequence of
only the not.first.false, which
describe each of infinitely.many,  and,
because not.first.false,
are true of each of infinitely.many.
The existence of 0 and x∪{y}
makes all the natural numbers explorable.
The existence of {y:p(y)}
makes ω explorable.

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.

∀ᑉᐜj ∃ᑉᐜk≠j: j<k  ⇔  ¬∃ᑉᐜk ∀ᑉᐜj≠k: j<k