Re: Small set Theory:final version.

zuhair wrote:
It is another way of defining "x is P_defined" .You can dispense with
it. I only wanted to mention it to add some clarification to this
concept.

If you've proven that it's equivalent, then save it as a theorem.

Ax.1: Description:AxEP ( x is P_defined )

'P' is now a variable that can range over formulas? That's not even a
first order axiom schemata. Or, IN THIS THEORY, you need to define 'is
a formula'. Or you have some kind of second order theory (and such that
I would suppose 'P' is not a formula but a predicate?).

Why Ax.1 states that for every x there exist a formula P in one free
variable such
that x is P_defined. I thought this was simple and clear. I just
thought the term
"formula in one free variable" is a clearily defined term.

In a first order theory you can't, in the object language quantify over
formulas (unless you have shown how to refer, from within the theory,
to the language of the theory, which is so remote from the present
context that I won't mention it again). You quantify 'EP'. You can't do
that. You can't use a quantifier of the object language ('E') to
quantify over formulas of the object language. But you can quantify
over formulas by using a quantifier of the meta-language, but you can't
insert it into the middle of an object language formula.

So, if by 'x' you mean the VARIABLE x, then you can say:

For every variable 'x', there exists a formula 'P' such that the
formula 'x is P_defined' is an axiom. Or something like that (I really
don't know if that's what you want, but at least it's coherent).

MoeBlee

.

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