Re: Replacement of Cardinality

On 8/16/2024 1:50 AM, Moebius wrote:

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:

On 8/15/2024 3:37 PM, Moebius wrote:

Am 15.08.2024 um 21:28 schrieb Jim Burns:

Defining ⅟𝔊 to be the smallest unit fraction
isn't a claim that ⅟𝔊 exists.

>
You need an existence proof [...]
BEFORE stating a/the proper definition.

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.

About things without an existence proof.

√2 is irrational.

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

Delete less.

⎛ Assume otherwise.

>
Nope. You clearly don't assume
 

√2 is rational ,

>
but:
>

⎜ Assume p₃,q₃ ∈ ℕ₁:  p₃⋅p₃ = 2⋅q₃⋅q₃

:

⎝ Contradiction.

I assume a statement equivalent to
⎜ Rational √2 exists.

________________________________
 *) 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.

Upthread is not a proof that real √2 exists.
Upthread is a proof that rational √2 not.exists,
which never mentions Dedekind completeness,
a never.mentioning which shows that,
whether or not real √2 exists,
rational √2 not.exists.
No, I am not proving that, here.
It makes my proof weaker and less focused, here.

 From (*) we get immediately: √2*√2 = 2.

You're complaining I didn't show you √2*√2 = 2.
Consider it left as an exercise for the reader.