Re: Small set Theory:final version.
- From: "zuhair" <zaljohar@xxxxxxxxx>
- Date: 31 Dec 2006 18:31:03 -0800
zuhair wrote:
MoeBlee wrote:
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:
The variable x here is a variable of sets.
For every variable 'x', there exists a formula 'P' such that the
formula 'x is P_defined' is an axiom.
But 'x is P_defined' is not a formula.
I mean x is P_defined is not the formula P. of course x is P-defined is
a formula.
Or something like that (I really
don't know if that's what you want, but at least it's coherent).
MoeBlee
I know you should be right about saying that but surely this is
something I cannot answer now. after I read the books you've mentioned
to me perhaps I can answer them.
But to me I see it understandable. What is the problem in saying that
for every set x there exist a formula P such that x is P_defined. You
say this is not a first order language? You are more aware from me
about this subject of course , but I see this as a formal limitation
rather than as a true limitation as regards the core of the subject,
since this axiom that I have put is understood in simple manner
therefore I don't know what is the importance of these formalities. Can
you give me a simple example to show me the consequences of my
bad formal language.
Zuhair
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