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From: Moebius <invalid@example.invalid>
Newsgroups: sci.logic,sci.math
Subject: Re: Replacement of Cardinality
Date: Fri, 16 Aug 2024 07:50:12 +0200
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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.