Re: For All x

On Tue, 3 Feb 2009 07:40:22 -0800 (PST), apoorv wrote:


Suppose we have a set theory with just one axiom:

Ez Ax ~x e z

"There is an empty set."

Would every set that contains 0 be a domain for a model of this
theory?

Yes - if, say, we interpret "e" as /e/ (at meta level). With other words,
the fact that our domain contains the empty set and that the empty set does
not contain any other set (i.e. any other element in the domain) together
with interpreting "e" as /e/ is what makes the statement/formula

Ez Ax ~x e z

true (in this interpretation).


In particular, would the truth value of Ax x=0 depend on the pre-
selected domain?

Well, you did not define "0" so far. So let me help you out here. Consider
"0" to be an _undefined_ notion of your system. And let's just consider the
set theory consisting of the single axiom:

Ax x !e 0.

"0 is empty."

Of course the truth value of

Ax(x = 0) (*)

Will depend on the interpretation (and hence domain) considered.

If the domain of a certain interpretation contains more that just one
element then (*) will be false (in this interpretation).


Herb


.

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