Why Extensionality is an axiom?
in [Logic]
|
Prev: All you have to do is c o m p r e h e n d this statement and diagonalisation falls apart
Next: Download new programs and english courses
From: MoeBlee on 12 Jun 2010 14:11 On Jun 12, 8:29 am, zuhair <zaljo...(a)gmail.com> wrote: > Hi all, > > I wonder why Extensionality is considered an axiom? One answer: extensionality characterizes sets in terms of membership. If extensionality doesn't hold for x and y, then one would likely conclude that x and y are not sets. > 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 You dropped a pair of parentheses. Should be: AxAy(Az(zex <-> zey) -> x=y) > Ac E!x Ay (y e x <-> (y e c & Phi)) [where 'x' does not occur fee in Phi] Yes, that proves extensionality. It's fine with me, but I could see that one might prefer a separate extensionality axiom in order to highlight it. By putting it separate and first, we immediately are clued in that the theory probably has sets as the intended subject matter. Unless someone mentions some special reason, to me it's merely a matter of style and it's fine either way. MoeBlee
From: George Greene on 12 Jun 2010 17:00 On Jun 12, 2:11 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > AxAy(Az(zex <-> zey) -> x=y) A bigger problem with Zuhair's axiom is the fact that <--> is its main connective. That makes it TWO axioms, not one. The --> version and the <-- version are different propositions. Another problem is the use of "=". Doing set theory in FOL *with* equality IS STUPID because equality is DEFINABLE in set theory. That's what extensionality IS, in set theory: THE DEFINITION of the symbol "=". x=y is just AN ABBREVIATION for Az[zex<->zey]. Definitions and axioms are NOT always the same kind of things! Properly, set theory is set in FOL withOUT equality, and one is simply trying to guarantee that sets with the same members "behave" indiscriminably, i.e., that Sets With The Same Members ARE Members of the SAME SETS. THE ACTUAL axiom of extensionality is Axy[ Az[zex<->zey] --> Az[ yez <-> xez ] ] Shrinking that is hardly worth the trouble. Again, = is an abbreviation, NOT ANYTHING WITH ANY LOGICAL CONTENT, so of course you can get "shorter" with it, BUT SO WHAT??
From: zuhair on 12 Jun 2010 17:52 On Jun 12, 4:00 pm, George Greene <gree...(a)email.unc.edu> wrote: > On Jun 12, 2:11 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > AxAy(Az(zex <-> zey) -> x=y) > > A bigger problem with Zuhair's axiom is the fact that <--> > is its main connective. That makes it TWO axioms, not one. > The --> version and the <-- version are different propositions. > > Another problem is the use of "=". > Doing set theory in FOL *with* equality IS STUPID because > equality is DEFINABLE in set theory. That's what extensionality IS, > in set theory: THE DEFINITION of the symbol "=". > x=y is just AN ABBREVIATION for Az[zex<->zey]. > Definitions and axioms are NOT always the same kind of things! > > Properly, set theory is set in FOL withOUT equality, > and one is simply trying to guarantee that sets with the same members > "behave" indiscriminably, i.e., that Sets With The Same Members > ARE Members of the SAME SETS. > > THE ACTUAL axiom of extensionality is > Axy[ Az[zex<->zey] --> Az[ yez <-> xez ] ] > > Shrinking that is hardly worth the trouble. > Again, = is an abbreviation, NOT ANYTHING WITH ANY LOGICAL CONTENT, > so of course you can get "shorter" with it, BUT SO WHAT?? The language that I was speaking about was first order logic with identity and membership. The axiom of Extensionality that I wrote is not my manufacture, it is the standard axiom written in this language. All my arguments was about writing matters in this particular language. And personally I see it much simpler to write set theory using FOL with identity and membership, than writing it in FOL with membership alone. According to your remarks, Quine must be a stupid man when he wrote NF in first order logic with identity. Most references write Z and related set theories in FOL(e,=), so all of them are doing what you call a stupid practice. Also you seem to forget identity theory, perhaps it is a stupid theories as well ha. When we use the symbol = as a primitive, we are saying that we are using the two schemes of identity theory as well. Regarding the bi-conditional, I agree with you it has the two directions, yet still matters would be shorter, than the standard manner. Zuhair
From: zuhair on 12 Jun 2010 18:07 On Jun 12, 4:00 pm, George Greene <gree...(a)email.unc.edu> wrote: > On Jun 12, 2:11 pm, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > AxAy(Az(zex <-> zey) -> x=y) > > A bigger problem with Zuhair's axiom is the fact that <--> > is its main connective. That makes it TWO axioms, not one. > The --> version and the <-- version are different propositions. > > Another problem is the use of "=". > Doing set theory in FOL *with* equality IS STUPID because > equality is DEFINABLE in set theory. That's what extensionality IS, > in set theory: THE DEFINITION of the symbol "=". > x=y is just AN ABBREVIATION for Az[zex<->zey]. > Definitions and axioms are NOT always the same kind of things! > > Properly, set theory is set in FOL withOUT equality, > and one is simply trying to guarantee that sets with the same members > "behave" indiscriminably, i.e., that Sets With The Same Members > ARE Members of the SAME SETS. > > THE ACTUAL axiom of extensionality is > Axy[ Az[zex<->zey] --> Az[ yez <-> xez ] ] > > Shrinking that is hardly worth the trouble. > Again, = is an abbreviation, NOT ANYTHING WITH ANY LOGICAL CONTENT, > so of course you can get "shorter" with it, BUT SO WHAT?? The language that I was speaking about was first order logic with identity and membership. The axiom of Extensionality that I wrote is not my manufacture, it is the standard axiom written in this language. All my arguments was about writing matters in this particular language. And personally I see it much simpler to write set theory using FOL with identity and membership, than writing it in FOL with membership alone. According to your remarks, Quine must be a stupid man when he wrote NF in first order logic with identity. Most references write Z and related set theories in FOL(e,=), so all of them are doing what you call a stupid practice. Also you seem to forget identity theory, perhaps it is a stupid theory as well ha. When we use the symbol = as a primitive, we are saying that we are using the two schemes of identity theory as well. Regarding the bi-conditional, I agree with you it has the two directions, yet still matters would be shorter, than the standard manner. Zuhair
From: Rock Brentwood on 21 Jun 2010 17:34
On Jun 12, 12:00 pm, zuhair <zaljo...(a)gmail.com> wrote: > On Jun 12, 11:43 am, George Greene <gree...(a)email.unc.edu> 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. > No I don't accept that. The uniqueness symbol is only shorthand > symbol, no symbol is added to the whole logical machinery. You probably will accept it, after you get through reading my reply. E!x.P(x) is shorthand for Ex.(P(x) & Ay.P(y) -> x e y)) OOPSEY! The "e" slipped back in under the sleeve of the shorthand. |