Re: ZFC++




All right, well let's establish what theory we're talking about first.
Give me a definition of the theory.

Though I already did, yet here is it:

Theory Y is the set of sentences entailed
(from first order logic with identity)by these axioms:

1) Extensionality: AxAy(x=y<->Az(zex<->zey)).

2) Universe: E!xAy(yex & ~Ez(~z=x & xez)).

Definition 1) x =T<-> Ay(yex & ~Ez(~z=x & xez)).

3) Regularity:
Ax(((Ez(zex))&~x=T)->Ey(yex & ~Ec(cey & cex))).

4) Schema of Global comprehension: if P is a formula in which x
doesn't occure free, then all closures of:

ExAy(yex<->(P(y)&Ez(~z=T & yez)))

are axioms.

Definition 2) x=V <-> Ay(yex<-> Ez(~z=T & yez)).

Definition 3) x is a proper class <-> (~x=T & ~Ey(~y=T & xey)).

Definition 4) x is a set <-> Ez(~z=T & xez)

Accordingly V is the proper class of all sets.

Definition 4) x=0 <-> Ay(~yex).

5) Pairing:AaeVAbeVExeVAyeV(yex<->(y=a or y=b)).

6) Union:AaExAy((yex<->Ez(zea&yez))&(Em(~m=T & aem)<->En(~n=T &
xen))).

7) Infinity:ENeV(0eN&(Ax(xeN->xU{x}eN))).

with 'U' and {x} having the usual definitions.

8) limitation of size:

Ax((Ez(~z=T & 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(~m=T & aem)<->
En(~n=T & & xen))).

10)Empty:ExeVAy(~(yex)).

/

You said that this theory is inconsistent, and it is trivial to prove
so.

Were is exactly the inconsistency?

Zuhair





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Relevant Pages

  • Re: Skolems Paradox
    ... sci.math_20050214.rtf:I suggest discarding all the non-logical axioms ... where in general theorems of ZFC minus regularity are theorems. ... Zermelo-Fraenkel Set Theory, is inconsistent, because of regularity. ... "p-adic integers") may well be infinite in precision and extent. ...
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  • Re: Skolems Paradox
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  • Re: ZFC++
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    (sci.math)
  • Re: ZFC++
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    (sci.math)
  • Re: Negating Quantifiers
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