Re: Replacement of Cardinality

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.)