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Topic: Interpreting Z and ZF
Replies: 2   Last Post: Apr 21, 2013 7:52 AM

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Zaljohar@gmail.com

Posts: 2,665
Registered: 6/29/07
Re: Interpreting Z and ZF
Posted: Apr 20, 2013 9:21 AM
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On Apr 19, 5:35 pm, Zuhair <zaljo...@gmail.com> wrote:
> Using theories that *interpret* Z; ZF can greatly shrink the
> axiomatics needed for proving the consistency of those theories, an
> old observation of course. The following couple of first order
> theories is a nice demonstration of that:
>
> C shall be used to denote the known subset relation.
>
> Theory 1 is the closure of all sentences following from:
>
> 1. Separation If phi is a formula in which y is free but not x,
> then all closures of:
>
> For all A Exist x For all y (y in x iff y C A & phi)
>
> are axioms.
>
> 2. Infinity.
>
> This interprets Z.
>
> Theory 2 is the closure of all sentences following from:
>
> (1) Replacement: If phi(x,y) is a formula in which x,y are free but
> not B,
> then all closures of:
>
> For all A  ([For all x C A Exist z for all y (Phi(x,y) -> y C z)]
> -> Exist B for all y (y in B iff Exist x C A (phi(x,y))))
>


Or we can use:

VA EB [(VyeB(ExeA.P)) & (VxeA (Ey.P -> EyeB.P))]

Where V signifies "for all"


> are axioms.
>
> (2) Infinity
>
> This interprets ZF.
>
> If one seeks to prove the consistency of Z or ZF, then proving the
> above theories would be sufficient, there is no need to try prove ALL
> the known axioms of those theories.
>
> Zuhair





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