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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.
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> > The following theory is equal to MK-Foundation-Choice
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> > Language: FOL(=,e)
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> > Define: Set(x) iff Ey. x e y
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> > Axioms: ID axioms+
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> > 1.Extensionality: (Az. z e x <-> z e y) -> x=y
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> > 2. Construction: if phi is a formula in which x is not free,
> > then (ExAy.y e x<->Set(y)&phi) is an axiom
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> > 3. Pairing: (Ay. y e x -> y=a or y=b) -> Set(x)
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> > 4. Size limitation
> > Set(x) <-> Ey. y is set sized & Azex(Emey(z<<m))
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> > where y is set sized iff Es. Set(s) & y =< s
> > and z<<m iff z =<m & AneTC(z).n =<m
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> > 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.
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> > y =< s iff Exist f. f:y-->s & f is injective.
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> > /
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> > 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.
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> 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?
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> 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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