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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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Posts: 2,665
Registered: 6/29/07
Re: An equivalent of MK-Foundation-Choice
Posted: Feb 21, 2013 7:39 AM
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On Feb 21, 2:01 am, Zuhair <> 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

An artificial fix is to define "set sized" in the following manner:

y is set sized iff [(E<3mey) or Es.Set(s)&y=<s]

where E<3mey means: there exist strictly less than 3 members in y.

[E<3mey] iff Ea,bAzey(z=a or z=b)

This obviates the need for axiomatizing pairing.

Anyhow this is just an artificial mix.


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