Why Extensionality is an axiom?
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From: Rock Brentwood on 21 Jun 2010 17:36 On Jun 21, 4:34 pm, Rock Brentwood <federation2...(a)netzero.com> wrote: > 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. Oopsey on my reply too. I meant x = y, not x e y. E!x.P(x) means Ex.(P(x) & Ay.P(y) -> x = y) I kept thinking you were using e for "=". Anyway ... the "=" slipped back in under the sleeve of the shorthand.
From: George Greene on 21 Jun 2010 19:14 On Jun 21, 5:36 pm, Rock Brentwood <federation2...(a)netzero.com> wrote: > Oopsey on my reply too. I meant x = y, not x e y. > > E!x.P(x) means Ex.(P(x) & Ay.P(y) -> x = y) > > I kept thinking you were using e for "=". Anyway ... the "=" slipped > back in under the sleeve of the shorthand. Exactly. Since = is what extensionality is trying to define, it would be circular to use = in the definition of a new quantifier need only for the definition of =. The RELEVANT point is that THE BASIC machinery of ZF is strong enough TO DEFINE, ELIMINABLY, BOTH E! *and* = !!!!
From: George Greene on 21 Jun 2010 19:21
On Jun 12, 6:07 pm, zuhair <zaljo...(a)gmail.com> wrote: > Most references write Z and related set theories in FOL(e,=), so all > of them are doing what you call a stupid practice. Yes, IT IS STUPID to do ZF in FOL(e,=), but it is not stupid enough to get worked up about. = IS DEFINABLE. Therefore, you are not hurting anything by speaking in that langauge, since you could always eliminate it if you wanted to. The subset predicate is not technically part of the language either, but people use it all the time, KNOWING THAT THEY CAN DEFINE IT AWAY. If I could use "c" for the sideways U that is usually written for "subset", then it is true that a=b <=df=> [ acb /\ bca ], so, again, = IS ELIMINABLE. The subset symbol itself is in turn eliminable, acb <=df=> Az[zea --> zeb]. SO THERE SIMPLY IS NO POINT to using = AS A PRIMITIVE in set theory. It is PURELY syntactic ICING on the first-order-language CAKE, which is THE (it is unique, up to orthography) first-order language WITH ONE binary predicate (and, maybe, one 0-ary function symbol or constant, 0, for the empty set). > Also you seem to forget identity theory, I don't "forget" it: I TOSS IT IN THE GARBAGE and I SPIT ON ITS GRAVE. It is ELIMINABLE from this logic, under this theory! > perhaps it is a stupid theory as well ha. No, that is not even PERHAPS the case. The point is that its entire VALUE, every consequence it entails, is achievable WITHOUT adopting it as a new primitive! > When we use the symbol = as a primitive, we are > saying that we are using the two schemes of identity theory as well. But in SET theory, WE NEVER say that, because WE CAN PROVE EVERYTHING that follows from those 2 schemata WITHOUT USING THEM. We DON'T NEED ANYthing from identity theory IN set theory, because set theory, in FOL withOUT equality, IS STRONG enough TO DEFINE equality! |