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

 Messages: [ Previous | Next ]
 Charlie-Boo Posts: 1,635 Registered: 2/27/06
Re: An equivalent of MK-Foundation-Choice
Posted: Feb 22, 2013 9:48 PM

On Feb 22, 3:38 pm, Zuhair <zaljo...@gmail.com> wrote:
> On Feb 22, 10:23 pm, Charlie-Boo <shymath...@gmail.com> wrote:
>
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> > On Feb 22, 5:14 am, Zuhair <zaljo...@gmail.com> wrote:
>
> > > On Feb 21, 11:10 pm, Charlie-Boo <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 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?
>
> > > > C-B
>
> > > > > 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?

>
> > C-B
>
> > > 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.

C-B

Date Subject Author
2/20/13 Zaljohar@gmail.com
2/21/13 Zaljohar@gmail.com
2/21/13 Charlie-Boo
2/22/13 Zaljohar@gmail.com
2/22/13 Charlie-Boo
2/22/13 Zaljohar@gmail.com
2/22/13 Charlie-Boo
2/22/13 Zaljohar@gmail.com
2/23/13 Charlie-Boo
2/23/13 Charlie-Boo
2/23/13 Zaljohar@gmail.com