
Re: An equivalent of MKFoundationChoice
Posted:
Feb 22, 2013 5:14 AM


On Feb 21, 11:10 pm, CharlieBoo <shymath...@gmail.com> wrote: > On Feb 20, 6:01 pm, Zuhair <zaljo...@gmail.com> wrote: > > > > > > > > > > > This is just a cute result. > > > The following theory is equal to MKFoundationChoice > > > 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? > > CB > > > > > > > > > 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

