
Re: An equivalent of MKFoundationChoice
Posted:
Feb 22, 2013 9:48 PM


On Feb 22, 3:38 pm, Zuhair <zaljo...@gmail.com> wrote: > On Feb 22, 10:23 pm, CharlieBoo <shymath...@gmail.com> wrote: > > > > > > > > > > > On Feb 22, 5:14 am, Zuhair <zaljo...@gmail.com> wrote: > > > > 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 > > > What has been proved? > > > > figuring it out before I send the whole written proof. > > > What are you waiting for? > > > I'll bet you $25 to your $1 that you don't supply a proof, payable via > > PayPal. Are we on? Only condition is you have to answer every > > question  no obfuscation, please. > > > It'd be well worth $25 if you have a proof of ZF consistency and I was > > one of the first to be able to give it. Has it been proven before? > > It seems people say "if ZF were consistent". Who has tried? Does > > Gödel's 2nd Theorem mean you'll do math that ZF can't? > > > CB > > > > Zuhair > > There is some confusion here. What I'm claiming is that IF we hold > that theory presented here to be consistent then MK minus foundation > minus choice would be consistent, and thus ZFC would be proved > consistent. I'm speaking about a relative consistency proof here. > > Zuhair
"Of course this theory PROVES the consistency of ZFC. Proofs had all been worked up in detail." This seems to be saying clearly that you have developed a proof that ZFC is consistent.
Now you're saying if one system is consistent then another is? Sure, if system X plus a few more axioms is consistent, then by definition system X is also consistent. Knowing that connection between 2 systems might signify nothing, depending on the relationship between the two systems.
CB

