Re: Small Set Theory,Updated.

zuhair wrote:
MoeBlee wrote:
zuhair wrote:
x is a set <-> EP(Ay(yex<->(P[y]&~y=x))).

would that be an incoherent statement?

First, is the expression well formed (or at least a rendering of a well
formed formula)? To answer that question, I would need to know the
formation rules of the language. The expression is not a rendering of a
well formed formula in a first order language. But I do know of
formation rules for a second order language that yield that the
expression is a rendering of a well formed formula. But I don't know
whether you intend for the expression to be in such a second order
language.

MoeBlee

Ok, Moe

let me see this then.

take the following statement:

Ay(yex<->(P[y]&~y=x))

I think this is an expression that is a well formed formula in a first
order language. One look to the axiom of separation in ZFC (which is a
well formed formula in first order language)

If we have a formal first order meta-theory, then the axiom schema of
separation is a well formed first order formula in the META-theory for,
say, Z set theory as an object theory, but the axiom schema of
separation is not itself a formula of the object theory. Each INSTANCE
of the axiom schema of separation is a well formed first order formula
of the object theory, but the axiom schema itself is in the language of
the meta-theory, not in the language of the object theory.

So if your 'P' is a 1-place predicate symbol of the object language,
then

Ay(yex <-> (Py & ~y=x)) is a formula of the object theory.

But if your 'P' is a meta-variable ranging over formulas of the object
language, then

Ay(yex <-> (P[y] & ~y=x)) is a schema, in the meta-language.

makes me think that the
formula I've wrote above is a first order language formula.

Instead of writing all of this formula which is a long formula, I made
an equivalent formula to it
to short it. and I call it " x is P_defined". so for example instead of
writing
Ex(Ay(yex<->(P[y]&~y=x))), you can write it as Ex(x is P_defined).
That's all. what is vague about that?

I didn't say that that particular construction is vague. However, I did
allude to saying that as you formulated certain definitional schema,
one might want to prove that the schema satisfies the criteria of
eliminability and non-creativity, but that I don't see any reason to
think eliminability and non-creativity would not be satisfied, so that
for working purposes I'm okay with working with your schema without
first going through the rigmarole of proving non-creativity and
eliminability.

What I do NOT permit is quantifying over formulas or over predicates
without either showing how to state 'is a formula' in the object
language or being explicit that we're in a second order language,
respectively.

I am sure that Ex(x is P_defined) is a first order language well formed
formula. Why not?
so is most of the complex formulae I wrote in this theory.

MoeBlee

.

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