Cardinals as Equivalence class?

Hi All,

In a previous Post, Moe Blee showed a proof that Frege's definition of
Cardinality is not in ZFC.

form what I remember his proof was like that.

Let c a set of all injectable sets to set x. were x is not empty

now let z e x.

then let Any v: v !e x replace z in x, then the resulting set is in c

y= ( x \ {z} ) U {v} -> y e c.

from this Moe Blee deduces that Uc is the set of all sets.

Therefore c is not a set is ZFC.
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My reply: I don't know till now how Uc is the set of all sets?

there is an intermediate step that I don't know in this proof.

Second: I personally have something which supports Moe Blee idea though
from another angle.

Since v is any set that is not in x, and since c is a set that is not
in x ( axiom of regularity ), then c itself can be v.

ie c itself can replace z in x and thus can be a member of x, then c is
a member of a member of c. which clearly viloates the axiom of
regularity.

ie instead of writting v: v !e x , since x in c then c !e x then , we
can take c itself.

y= ( x \ {z} ) U {c} -> y e c which violates the axiom of regularity,
since c is a member of y.

so c is not a set is ZFC, But this can be corrected is we say that c is
a PROPER CLASS
since proper classes cannot be members of other sets, ie do not have
supersets.

I don't know whether if we assume that c is a proper class will also
work in the same way on Moe Blee's proof, since Uc ( I think this is
the set union of c, ie the superset of c ) do not exist for
c if c is a proper class.

so an Equivalence class of sets , under equivalence relation
"bijection" is in reality a proper class and it would be better writtin
as.

Cardinal <-> Equivalence proper class of sets , under equivalence
relation "bijection".

Similar thing applies to the definition of ordinals as equivalence
classes, it should be like below:

Ordinal <-> Equivalence proper class of sets , under equivalence
relation "order isomorphism".

Zuhair

.