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Topic: An equivalent of MK-Foundation-Choice
Replies: 10   Last Post: Feb 23, 2013 11:20 PM

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Zaljohar@gmail.com

Posts: 2,665
Registered: 6/29/07
Re: An equivalent of MK-Foundation-Choice
Posted: Feb 22, 2013 5:14 AM
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On Feb 21, 11:10 pm, Charlie-Boo <shymath...@gmail.com> wrote:
> On Feb 20, 6:01 pm, Zuhair <zaljo...@gmail.com> wrote:
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> > This is just a cute result.
>
> > The following theory is equal to MK-Foundation-Choice
>
> > Language: FOL(=,e)
>
> > Define: Set(x) iff Ey. x e y
>
> > Axioms: ID axioms+
>
> > 1.Extensionality: (Az. z e x <-> z e y) -> x=y
>
> > 2. Construction: if phi is a formula in which x is not free,
> > then (ExAy.y e x<->Set(y)&phi) is an axiom

>
> > 3. Pairing: (Ay. y e x -> y=a or y=b) -> Set(x)
>
> > 4. Size limitation
> > Set(x) <-> Ey. y is set sized & Azex(Emey(z<<m))

>
> > where y is set sized iff Es. Set(s) & y =< s
> > and z<<m iff z =<m & AneTC(z).n =<m

>
> > TC(z) stands for 'transitive closure of z' defined in the usual manner
> > as the minimal transitive class having z as subclass of; transitive of
> > course defined as a class having all its members as subsets of it.

>
> > y =< s iff Exist f. f:y-->s & f is injective.
>
> > /
>
> > Of course this theory PROVES the consistency of ZFC.
> > Proofs had all been worked up in detail. It is an enjoying experience
> > to try figure them out.

>
> Why don't you supply your proof of ZFC consistency in detail then?  If
> that is too much, then why not give it for 2 potentially conflicting
> axioms in detail, as I suggested recently?
>
> Why waste space with unsubstantiated claims of grandeur?
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> C-B
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> > Zuhair

Of course I'll supply them in DETAIL. There is no grandeur, nothing
like that. That matter has been PROVED. I just wanted some to enjoy
figuring it out before I send the whole written proof.

Zuhair



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