Why Extensionality is an axiom?
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From: zuhair on 12 Jun 2010 11:29 Hi all, I wonder why Extensionality is considered an axiom? What I mean by Extensionality is the following sentence in FOL with identity"=",and membership"e": Ax Ay Az(z e x <-> z e y) -> x=y Now this sentence can be a theorem of a theory that has exactly all axioms of Z with a very slight modification on the formula in Separation to the following: Ac E!x Ay (y e x <-> (y e c & Phi)) Proof: take Phi <-> y e c substitute in Separation, then we'll have Ac E!x Ay (y e x <-> y e c) which entails Extensionality. QED It seems to me that this approach is quite simpler than the standard one, in which Extensionality is included as an axiom. So why Extensionality is included among the list of axioms of Z, if only adding ONE symbol to Separation, does the job? Zuhair
From: George Greene on 12 Jun 2010 12:43 On Jun 12, 11:29 am, zuhair <zaljo...(a)gmail.com> wrote: > So why Extensionality is included among the list of axioms of Z, if > only adding ONE symbol to Separation, does the job? Probably because the "one symbol" has to be added to the whole logical machinery, that's why. You are introducing a whole new quantifier. Adding an axiom to a theory is arguably easier than changing the whole logic.
From: George Greene on 12 Jun 2010 12:45 On Jun 12, 11:29 am, zuhair <zaljo...(a)gmail.com> wrote: > It seems to me that this approach is quite simpler than the standard > one, in which Extensionality is included as an axiom. > > So why Extensionality is included among the list of axioms of Z, if > only adding ONE symbol to Separation, does the job? Z is old. Everybody uses ZF instead. You should try throwing AWAY Separation and Pairing and seeing if you can do this from Replacement.
From: zuhair on 12 Jun 2010 12:59 On Jun 12, 11:45 am, George Greene <gree...(a)email.unc.edu> wrote: > On Jun 12, 11:29 am, zuhair <zaljo...(a)gmail.com> wrote: > > > It seems to me that this approach is quite simpler than the standard > > one, in which Extensionality is included as an axiom. > > > So why Extensionality is included among the list of axioms of Z, if > > only adding ONE symbol to Separation, does the job? > > Z is old. > Everybody uses ZF instead. > You should try throwing AWAY Separation and Pairing and seeing if > you can do this from Replacement. Yes, of course the same thing applies with Replacement, just add the uniqueness symbol that's it. Zuhair
From: zuhair on 12 Jun 2010 13:00
On Jun 12, 11:43 am, George Greene <gree...(a)email.unc.edu> wrote: > On Jun 12, 11:29 am, zuhair <zaljo...(a)gmail.com> wrote: > > > So why Extensionality is included among the list of axioms of Z, if > > only adding ONE symbol to Separation, does the job? > > Probably because the "one symbol" has to be added to the whole > logical machinery, that's why. You are introducing a whole new > quantifier. > Adding an axiom to a theory is arguably easier than changing the whole > logic. No I don't accept that. The uniqueness symbol is only shorthand symbol, no symbol is added to the whole logical machinery. Zuhair |