For All x

When we use the quantifier 'For All x' the variable x can range over
all values in the domain under consideration.Does it mean that we know
beforehand what is in the domain and what is not?
As an example, consider a set theory with only two of the usual
axioms :
1) Ez:Ax~x e z ( existence of 0 )
2)Az Ey Ax x e y <-> x e z & 0 e x (specific instance of separation)

This system, at the object level, is agnostic about the existence of
a successor of 0.

If z in axiom 2 can range only over the sets explicitly guaranteed
to
exist by the axiom system without reference to the domain ,then 2)
reduces to
Ey Ax xe y <-> x e 0 & 0 e x , so that y=0.The two axioms , by
themselves can
assert the existence of only the set 0

If the domain is specified beforehand to contain 0, &{S0} , then 2)
can assert the existence of
both 0 (z=0) and {S0} (z={S0})

which of these ways of interpreting 'for all x' , namely x ranging
over sets whose existence
can be proved at the object level, OR x ranging over a pre-specified
domain --is the correct one? shall appreciate a clarification.
-apoorv





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