Re: Simple enough for every reader?

On 2025-06-13 13:49:25 +0000, WM said:

On 13.06.2025 12:02, Mikko wrote:

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.

 It is part of the proof because it specifies the set which the proof is based upon.

Doesn't really matter because the "proof" does not end with an interesting
concousion.

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 is always concerning individuals only.

Therefore "as individuals" is meaningless.

∀n ∈ ℕ: P(n) specifies that every natural number n as an individual has the property P.

That is an ordinary quantification, not a quantification as individuals.
--
Mikko