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Date: Tue, 04 Jun 24 14:10:43 +0000
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From: WM <wolfgang.mueckenheim@tha.de>
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Le 04/06/2024 à 04:07, Jim Burns a écrit :
> On 6/3/2024 3:50 PM, WM wrote:

> Assumption (2.) describes
> objects in our familiar arithmetic.

That is true because our familiar arithmetic is based upon potential 
infinity.
Only in finished infinity dark numbers are required. 

> For each n in ℕ⁺
> n countable.to from.0
> n+1 is countable.to from.n
> n+1 is countable.to from.0 through.n
> n+1 is in ℕ⁺
> n is not larger than all numbers in ℕ⁺

And in particular all n have ℵo dark successors.

> 𝔼 is the subset of even numbers in ℕ⁺
> 𝕆 is the subset of odd numbers in ℕ⁺
> 
> ⎛ There is no number in 𝔼 larger than
> ⎝ all numbers in 𝕆
> 
> ⎛ There is no number in 𝕆 larger than
> ⎝ all numbers in 𝔼

Dark numbers do not unveil their mystery.

But from NUF we can prove that a first one must exist although we cannot 
find it.

>> Only one of the two complementary sets
>> can and must contain the first point.
> 
> Why?
> Responding "Logic" or "Mathematics" is dodging.

But it is true. Like Bob's survival.

Regards, WM