ZFC++
- From: "zuhair" <zaljohar@xxxxxxxxx>
- Date: 28 Feb 2007 19:21:22 -0800
Hi all,
ZFC+Proper classes+Ur elements'ZFC++':is the set of sentences entailed
(from first order logic with identity)by these axioms:
1) Extensionality: AxAy(x=y<->Az(zex<->zey)).
2) ur-element:Ex(Ay(yex<->y=x)).
Definition 1) x is a ur-element<-> Ay(yex<->y=x).
3) Regularity:
Ax((Ez(zex))->Ey(yex & ~Ec(cey & cex & ~c=y))).
4) Schema of Global comprehension: if P is a formula in which x is not
free then all closures of:
ExAy(yex<->(P(y)&Ez(yez)))
are axioms.
Definition 2) x=V <-> Ay(yex<->(y=y & Ez(yez))).
Definition 3) x is a proper class <-> ~Ey(xey).
Accordingly V is a proper class.
Definition 4) x=0 <-> Ay(~yex).
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))).
with 'U' and {x} having the usual definitions.
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)).
Definition 5) x=K<-> Ay(yex<->y is a ur-element)
so K has all ur-elements as its members.
10)ur-multiplicity:
Ax((Az(zex->zeK)&Eu(xeu))->Ey(y is a ur-element & ~yex)).
Accordingly K is a proper class.
/
Zuhair
.
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