Re: ZFC+
- From: "zuhair" <zaljohar@xxxxxxxxx>
- Date: 28 Feb 2007 04:24:07 -0800
On Feb 28, 2:45 am, "Rupert" <rupertmccal...@xxxxxxxxx> wrote:
On Feb 28, 2:55 pm, "zuhair" <zaljo...@xxxxxxxxx> wrote:
On Feb 27, 10:09 pm, "Rupert" <rupertmccal...@xxxxxxxxx> wrote:
On Feb 28, 7:25 am, "zuhair" <zaljo...@xxxxxxxxx> wrote:
Yep, I'm pretty sure. We could interpret the ur-elements as the
ordinals, for example.
Can you clarify this point, with some detail.
How can you exactly interpret this theory with NGB.
My original proposal to identify the ur-elements with the ordinals
won't work. It's not as simple as I thought. I'll have a bit more of a
think about it.
Yea, I guessed that. This theory is very complicated. Concentrate on
Regularity, because I think it is actually erronous, anyhow.
I have no doubt that the theory can be interpreted in NBG, it's just
that we'll need quite a complicated translation scheme.
ah, ic. when you say interpreted in NBG,what you are exactly hinting
at. Do you mean that this theory is entailed by NBG,or do you mean it
is equi-interpretable with NBG.
It's equi-interpretable with a subtheory of NBG with global choice.
which subetheory of NBG it is equi-interpretable to.
Can you give me details.
Zuhair
(We need global choice in order get the axiom of limitation of size).
The axiom of regularity here is different.this theory allows sets that
are not allowable in NBG. Can you prove the consistency of this theory
from NBG.
Yes, if NBG with global choice is consistent then your theory is
consistent.
Zuhair
I would have agreed with you about the theory of ZFC+, i.e the one
without ur-elements. But even ZFC+ , if you say it is interpretable
with NBG, my question is NBG is finitelly axiomatized, while this
theory is infinintelly axiomatized.
Can they be interpretable?
Yes, there's no problem. And I'm not saying they're equi-
interpretable, I'm just saying your theory can be interpreted in NBG.
I might need the axiom of global choice as well.
Second I managed to reach into another method of incorporating the ur-
elements to this theory.
see this theory.
ZFC++,
Language: 1st order logic
Primitives: e, =
Definition1)
x is a set <-> (Ay(~(yex)) or Ey(yex)).
Definition 2)
x is a ur-element <-> ((~( x is a set)) & Ez(xez)).
1) Extensionality of sets :
Ax:x is a set Ay:y is a set (x=y<->(Az(zex<->zey)).
2) Identity of ur-elements :
Ax: x is a ur-element Ay: y is a ur-element ( x=y <->
Az:z is a set ( xez <-> yez ).
3) Regularity:
Ax((Ez(zex))->Ey(yex & ~Ec(cey & cex))).
4) Global comprehension: if P is a formula in first order logic in
which x is not free, or P uses solitary primitive
predicates(predicates that doesn't use the
membership 'e' in them, and consists of only one primitive predicate
that is postivelly stated, i.e does't have '~' in it,like
Car(x),Man(x),Sun(x),... )then the sentence:
ExAy(yex<->(P(y)&Ez(yez)))
is an axiom.
Definition 3) x=V <-> Ay(yex<->(y=y & Ez(yez))).
Definition 4) x is a proper class(large set) <-> (x is a set &
~Ey(xey)).
Definition 5) x is a small set <-> (x is a set & Ey(xey)).
Accordingly V is the proper class(large set) of all small sets and ur-
elements.
5) Pairing:AaeVAbeVExeVAyeV(yex<->(y=a or y=b)).
6) Union:AaExAy((yex<->Ez(zea&yez))&(Em(aem)<->En(xen))).
7) Infinity:ENeV(0eN&(Ax(xeN->xU{x}eN))).
8) limitation of size:
Ax((Ez(xez)) <-> x is subnumerous to V).
x is subnumerous to V <-> Af((f:x->V) ->(f is injective & ~ f is
surjective)).
9)Power: AaExAyeV((yex<->Az(zey->zea)&(Em(aem)<->En(xen)).
10)Primitive predicate:
if P0 is a solitary primitive predicate(see above). then the
sentence:
Ax(P0(x)-> x is a ur-element)
is an axiom.
I think this is the most versatile theory, having small sets
large sets (proper classes) and ur-elements.
This theory clearly extends imperical predicates like Man(x)
Sun(x),etc..., into it as ur-elements.
Interesting theorums especially about ur-elements can spring from this
theory. like Ax(x is a ur-element->P(x)=0)
Also Ax(x is a ur-element -> Ux=0).
I think this is stronger than the first one , in which I tried to
define ur-elements in as singlton sets in themselfs.
This theory by far much stronger than ZFC, which only allows for the
existance of small sets only. All axioms and theorums
of ZFC, are in this theory. But the converse is not true.
Even ZFC+ alone is much stronger than ZFC.
I think that this would come up to be much more fascinating theory
than ZFC is.
Zuhair- Hide quoted text -
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