Re: For All x

On Mar 12, 11:35 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Mar 12, 3:13 am, apoorv <sudhir...@xxxxxxxxxxx> wrote:

On Mar 10, 12:55 am, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:> On Mar 8, 11:48 pm, apoorv <sudhir...@xxxxxxxxxxx> wrote:

We add extensionality

That will facilitate things. And you now can define '1'. But it still
is not enough for your claim that DeD.

since by the axioms ,a (unique) universal set exists, i am unable to
see the basis for your assertion that the theory does not prove D e
D.

I've explained it half a dozen times already. 'D' is defined in the
the meta-theory, not in your object theory.

Yes, your theory proves Ex xex. But that says NOTHING about D, which
is something discussed in a DIFFERENT theory.
i think we are covering the same ground again ;D is the unique
universal set in the theory and clearly D e D. I am unable to see
where is this different theory coming from.
You have not yet proven even that there is a number 1, let alone
anything of greater mathematical interest.

I am really curious to know why you believe that this theory 'proves'
the existence of any 'object' any less than ZFC

I'll believe it proves all the existence theorems of ZFC when you
prove that it does. Meanwhile, since your theory is weaker than ZFC,
it's pretty ease to see that it doesn't prove all of existence
theorems of ZFC.

There is no attempt to show that this theory proves all the existence
theorems of ZFC. Indeed, the main idea is to set up a theory which
allows for the
exisrence of just one infinite set and no more.

But previously you referred to all kinds of things as if defined in
your object theory. Please make up your mind.
I think you are deliberately obfuscating and misinterpreting. since
you have been in this thread continuously , it is hard to believe that
you missed that the theory
was designed to allow for the existence of one infinite set and no
more.
My own sense is that you are so mesmerized by this garden of the
transfinite and your own (believed) felicity with some of the
symbolism that you are unable to see or would not see any other
perspective. Pleasse wake up!
And now how do you define 'infinite' in your object theory and prove
Ex x is infinite in your object theory?
A simple definition is obvious. it is the only set that is its own
successor.
I'm not sure I understand your question. But if a declared meta-theory
R for a theory T is such that "Ax x is set" (where, by definition, 'x
is a set <-> (x = the_y(Az ~zey) v Ezy(zexey))') then, perforce, any
domain of a model is a set.

If I read your definition right, you have not excluded the possibilty
x e x
for a set.

It depends on T.
you specified T by "Ax x is aset " and defined 'x is a set'. That is
adequate for the possibilty x ex.




Generally in mathematical logic, the DEFINITION of a model includes
that the domain is a set; and the theory in which mathematical logic
and model theory is formalized is presumed to be some kind of set
theory. However, of course, one could adopt a class theory such as NBG
as the theory in which to formalize, or even some other kind of formal
foundational theory; and one could adopt a different defintion of
'model' so that proper classes could be domains. Yet, this is not
without some difficulty at least. (See, for example, remarks in Chang
& Keisler, where Morse class theory is used as the official meta-
theory.)

So, if you have some alternative meta-theory, you should declare it so
that we can evaluate your claims in that regard.

 The theory defines its own domain and need not call upon any other
theory to supply a domain for a meaningful interpretation of its
variables.

Bull. Pure unsupported assertion.
I reciprocate the sentiments.
Nonetheless,
if the domain has to be a set in a more conventionally formulated
theory, we
could consider ZFc-regularity +~regularity , with the ~regularity
being incorporated in the theory

Here, by 'the theory', we're talking about the META-theory.

by explicitly asserting the existence
D defined by x e D iff (x is a finite ordinal or x =D)

"ASSERTING existence". I.e., taking as an axiom for the META-theory.
Fine.

Now specify a 2-place relation R on D such that when the universal
quantifier of your object language maps to D, and 'e' (from your
object language) maps to R, we have that every axiom of your OBJECT
theory is true by the just mentioned mapping (i.e., the just mentioned
structure for the object language).
i would think that x R y <--> x e y adequately meets the requirement.
-apoorv
.

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