Re: For All x

On Feb 2, 5:03 pm, William Elliot <ma...@xxxxxxxxxxxxxxxx> wrote:
On Mon, 2 Feb 2009, apoorv wrote:
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?

In ZF, there is no domain given nor does any FOL have a domain.
Domains are a part of model theory and will vary from model to
model for the same FOL theory.

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)

That seems wrong. I'll presume you mean
Az Ey Ax (x e y <-> x e z & 0 e x)

Don't you also need, a = b iff for all x, (x in a iff x in b) ?

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

Again you repeat the same mistake.
  Ey Ax (xe y <-> x e 0 & 0 e x)

It is obvious that y = 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})

If you add the axiom, for all x, x = 0;
I don't know what &{S0} means;
the rest is vague beyond rescue.

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

The two axioms have a model with a domain of one element.
This means that you cannot prove the existence of a set
from the axioms, other than 0.

I was not able to convey my question precisely. One more attempt.
Suppose we have a set theory with just one axiom:
Ez Ax ~x e z [i.e there is a null set 0]
Would every set that contains 0 be a domain for a model of this
theory?
In particular, would the truth value of Ax x=0 depend on the pre-
selected domain?
-apoorv
.

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