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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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Charlie-Boo

Posts: 1,588
Registered: 2/27/06
Re: An equivalent of MK-Foundation-Choice
Posted: Feb 23, 2013 7:16 AM
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On Feb 20, 6:01 pm, Zuhair <zaljo...@gmail.com> wrote:
> This is just a cute result.

Is that an attempt to brag in the context of being modest but not
really by calling it cute?

just = only = modest
&
cute = nothing significant, just looks like a precious little baby
but
result = new discovery in the history of mathematics = very
significant

Which is it - modesty or delusions of grandeur?

C-B

> 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.
>
> Zuhair





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