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Topic: Hereditary size notion proving consistency of ZF
Replies: 1   Last Post: Jan 20, 2013 9:34 AM

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Posts: 2,665
Registered: 6/29/07
Re: Hereditary size notion proving consistency of ZF
Posted: Jan 20, 2013 9:34 AM
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On Jan 19, 9:20 pm, Zuhair <> wrote:
> I think that ZF can be proved consistent relative to consistency of
> the following theory defined in the same language of ZF with the
> following axioms:
> Define: Set(x) iff Exist y. x in y
> [1] Extensionality: as in Z
> [2] Comprehension: If phi is a formula in which x is not free, then
> (Exist x. for all y. y in x iff Set(y) & phi) is an axiom.
> [3] Pairing: (for all y. y in x -> y=a or y=b) -> Set(x)
> [4] Size: [Set(x) & y < x & for all m in z (m << Uy)] -> Set(z)

If we remove the condition y < x and replace Uy with x then this would
be sufficient to prove the consistency of Z.

> Definitions:
> y < x is defined as: Exist s,f: s subset_of x & f:s --> y & f is
> surjective.
> y << x is defined as: y < x & (for all z in TC(y). z < x)
> TC(y)=x iff [for all z. z in x iff (for all s. y subset_of s & s is
> transitive -> z in s)]
> Uy=x iff [for all z. z in x iff Exist k. k in y & z in k]
> /

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