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Re: IMPROVED AXIOM OF REGULARITY... <X1 e X2 e X3 .. e Xn e X1>
Posted:
May 11, 2012 2:34 PM
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On May 12, 2:33 am, MoeBlee <modem...@gmail.com> wrote:
> On May 11, 11:24 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > > My proof (including any lemma upon which the proof depends) makes no
> > > use nor mention of powersets, the power set axiom, or V (I surmise you
> > > mean 'V' as the class of all sets).
>
> > Yes!
>
> Yes, right. The proof makes no use of power sets, the power set axiom,
> or of the class of all sets. (And even if the power set axiom were
> used, still the proof would be in Z set theory, perforce in ZFC).
>
> > x is not an element of V
>
> > x e P(V)
>
> You're confused. V is not invoked in the proof. Really, I don't see
> why you won't read an introductory textbook in set theory.
>
> MoeBlee
You really do not see the difference between the equivalent formula
Ax(~x=0 -> Em(mex & x/\m = 0))
and
A(X) A(Z) (X t Z) -> ~(Z e X)
X is a theorem, x is a set of theorems.
The purpose of set theory axioms is not to prove that set theory
works, they are supposed to be utilised to instantiate new formula.
If you had 10^7 sets in your theory and added 1 more:
a) my set level AOR-ii would take O(n log(n)) checks
to instantiate 1 more formula.
b) the powerset level ZFC-AOR would take |P(10^7)| checks
to instantiate 1 more formula.
e.g.
V = { SET1, SET2, SET3 }
<=>
V = { {SET8}, {SET3,SET4}, {SET1,SET2,SET5} }
i.e.
SET1 = {8}
SET2 = {3,4}
SET3 = {1,2,5}
P(V) = {
{},
{{8}},
{{3,4}},
{{1,2,5}},
{{8},{3,4}},
{{8},{1,2,5}},
{{3,4},{1,2,5}}, **
{{8},(3,4),{1,2,5}}
}
NOW you can invoke ZFC-AOR on P(V)
Ax(~x=0 -> Em(mex & x/\m = 0))
WHEN
x = {{3,4},{1,2,5}}
ie. x = {{SET3, SET4}, {SET1, SET2, SET5}}
ie. x = {SET2, SET3}
m=SET2
x/\m = x/\SET2 = x/\{SET3,SET4} = {SET3,SET4} =/= 0
m=SET3
x/\m = x/\SET3 = x/\{SET1,SET2,SET5} = {SET1,SET2,SET5} =/= 0
AOR is not satisfied... eventually!
***********************************
Compare that to a simple TRANSITIVE RELATION and AOR-II
-------AXIOM OF REGULARITY II------
AXIOM OF TRANSITIVITY
A(X) A(Z) X t Z <-> (X e Z) v E(Y) (X e Y) ^ (Y t Z)
AXIOM OF REGULARITY
A(X) A(Z) (X t Z) -> ~(Z e X)
-----------------------------------
Herc
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