Re: Replacement of Cardinality

Am 16.08.2024 um 07:05 schrieb Jim Burns:

On 8/15/2024 7:16 PM, Moebius wrote:

Am 16.08.2024 um 00:51 schrieb Jim Burns:

 

Half or more of my proofs to WM say
"Assume otherwise... However... Contradiction."

>
Yeah, to be precise a proof by contradiction
assumes a STATEMENT/CLAIM.

 Many times, a false existence claim.

Right.

√2 is irrational.

This statement is just nonsense, _if_ "√2" is not already defined.*)

⎛ Assume otherwise.

Nope. You clearly don't assume
       √2 is rational ,
but:

⎜ Assume p₃,q₃ ∈ ℕ₁:  p₃⋅p₃ = 2⋅q₃⋅q₃

:

⎝ Contradiction.

________________________________
*) Of course, _if_ we already have introduced the real numbers (i.e. IR) we may define
       √2 = the real number x such that x*x = 2 ,    (*)
_after_ we have shown that that there is exactly one x e IR such that x*x = 2.
 From (*) we get immediately: √2*√2 = 2.