Re: Simple enough for every reader?

On 2025-06-12 09:44:06 +0000, WM said:

On 12.06.2025 10:00, Mikko wrote:

On 2025-06-11 11:38:52 +0000, WM said:

 

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.

 It does not show that. The "proof" does not even mention definability.

 "this proof for all definable numbers n"

That is a part of the sentence that mentions the proof, not a part of
the proof itself.

The conclusion follows from the second sentence alone when n is understood
to be universally quantified, so the first sentence is not needed and should
not be there.

 It is there. Only definablenumbers can be quantified as individuals.

Quantification is never "as individuals". It mevery specifies the type
of the quantification (like "every", "some", or "none"), introduces a
bound variable, and specifies the range of values for that variable.
--
Mikko