Re: Replacement of Cardinality

Am 24.08.2024 um 00:04 schrieb Jim Burns:

x = 1/0  :⇔  0⋅x = 1

Using a slight variant of your "definition" it's easy to show why your approach (concerning definitions) is not tenable in math:
Def.: x = 1/0  :⇔  x ∈ IR & 0⋅x = 1
Now from identity theory (say in the context of FOPL=) we get
       Ax(x = x)
and hence (by specialisiation, AE):
       1/0 = 1/0 .
[We may do this, because "1/0" is now - after your definition - a term in our system.]
 From this we get (by EI):
       Ex(x = 1/0)
and from this with our "definition":
       Ex(x ∈ IR & 0⋅x = 1) .
Not a desirable result.
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Again, that's why we have to prove an existence claim
           Ex(Px)
before stating/using the defintion
           x = c :<-> Px .
Actually, in addition to this proof we also need a proof of "uniquines" before ...