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From: Moebius <invalid@example.invalid>
Newsgroups: sci.logic,sci.math
Subject: Re: Replacement of Cardinality
Date: Thu, 15 Aug 2024 23:43:44 +0200
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Am 15.08.2024 um 23:36 schrieb Moebius:
> Am 15.08.2024 um 20:36 schrieb Jim Burns:
> 
>> Translate ¬∃ᴿx(x = 1/0) to
> 
> There is nothing to translate. "¬∃ᴿx = 1/0" is just a meaningless 
> expression, because "1/0" is a undefined (non-denoting) term/name.
> 
>> ¬∃ᴿx: 0⋅x = 1
> 
> Now this is a meaningful statement.
> 
>> Prove that.
> 
> Indeed! :-)
> 
> For this we might assume
> 
>       ∃ᴿx: 0⋅x = 1
> 
> and try to derive a contradiction from this assumption.
> 
> ->Proof by contradiction (RRA).

Hint: And BECAUSE we can prove:

        ¬∃x(x e IR & 0⋅x = 1)

we CAN'T define "1/0" the following way:

         1/0 := the x e IR such that 0⋅x = 1 .

Nuff said.

(It should be clear that the function x |-> 1/x is not defined for x = 
0, hence even we have defined x |-> 1/x (for x e IR, x =/= 0), we may 
not write "1/0", based on THIS definition.)