Re: Simple enough for every reader?

On 11.06.2025 10:01, Mikko wrote:

On 2025-06-03 13:26:11 +0000, WM said:

If there are more than finitely many numbers following upon every definable number as successors, then every definable number has more successors than predecessors.

 Provable.
 

All can be removed collectively. ℕ \ {1, 2, 3, ...} = { }.

 The word "collectively" is not a well defined mathematical term.

It is well defined because collectively means the whole set.

It can
be used in informal presentation but not in a rigorous proof.
 

If all could be removed as individuals, then a last one would be removed. Contradiction.

 Likewise, "as individuals" has no well defined mathematical meaning
so cannot be used in a rigorous proofs.

"As an element" has a well defined meaning. But it is useless to be well-defined in a wrong theory like ZF where rigorous proofs fail to establish the existence of dark numbers.

 

This is a proof of dark natural numbers.

 It is not a proof of anything.
 

Outside of ZF in correct mathematics this proof for all definable numbers n
|ℕ \ {1}| = ℵo.
|ℕ \ {m ∈ ℕ | m < n}| = ℵo
==> |ℕ \ {m ∈ ℕ | m < n+1}| = ℵo.
shows that is impossible to extend definability to all natural numbers with none remaining undefined.
Regards, WM