The set of All sets
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From: zuhair on 15 Dec 2005 04:13 MoeBlee wrote: > zuhair wrote: > > axiom of extensionality > > states that:-Two "SETS" are the same if and only if they have the same > > elements. > > No, in a formal first order language, the axiom of extensionality is: > > AxAy(Az(z e x <-> z e y) -> x = y) > > There is no mention of sets there. Informally, to convey the sense of > the axiom, one may say, "If sets have the same members then the sets > are equal", but this is not required. We do not need to use the word > 'set' to do set theory, unless we need to distinguish among sets, > individuals, and classes, which ZF does not. > > > About predicate logic you are using , non of ZFC axioms mentions that > > explicitely. > > The axioms are IN a first order language' the axioms don't need to > mention that they are in the language. That is what is mentioned in the > meta-theory. In an informal exposition of the theory, one might not > mention that the theory is in a first order language, and we use the > word 'sets', but when we talk about the actual formal mathematics, we > talk about a first order theory that does not have a primitive > predicate symbol for 'is a set', though a predicate symbol for 'is a > set' can be defined. > > Why don't you study a book on mathematical logic and then one on set > theory rather than remain completely and continually uniformed and > confused? > > MoeBlee So you think that hiding behind first order language would make you escape the cyclicality of sets, but it doesn't ! even that doesn't solve the issue! tell me my dear what is "e" ? z e x means z belongs to x , so z is a member of the set x, if x was not a set then z cannot belong to x and z e x would be nothing but handwaving. we need to know how we can define x as a set for the first order language z e x to have any meaning! In reality I do really doubt that e has a definition, I think it is undefined , in reality neither sets nor belonging to sets is defined by ZFC, take a look at the axioms and tell me where is the definition of x and y and z and e? There is non! Still the same question is there : What is a set? more what is "set membership"? Zuhair
From: MoeBlee on 15 Dec 2005 20:59 zuhair wrote: > tell me my > dear what is "e" ? 'e' is undefined. Aside from '=', 'e' is the only undefined predicate symbol of Z set theory (and '=' can be defined by 'e'). Within a few steps, we can define '0': 0 = the unique x such that Ay ~y e x. Then we can define 'is a set': x is a set <-> x = 0 v Ey y e x. If that leads you to complain that 'e' is undefined, then I can't help you. Either you have primitives for the language, or you develop mathematics in some other way, without at least one primitive predicate symbol (or operation symbol). Since I don't know a way to develop mathematics without primitives, I find it quite reasonable - remarkable and beautiful - that set theory can do the entire job with just one. MoeBlee
From: zuhair on 19 Dec 2005 06:42
MoeBlee wrote: > zuhair wrote: > > tell me my > > dear what is "e" ? > > 'e' is undefined. Aside from '=', 'e' is the only undefined predicate > symbol of Z set theory (and '=' can be defined by 'e'). Within a few > steps, we can define '0': 0 = the unique x such that Ay ~y e x. Then we > can define 'is a set': x is a set <-> x = 0 v Ey y e x. > > If that leads you to complain that 'e' is undefined, then I can't help > you. Either you have primitives for the language, or you develop > mathematics in some other way, without at least one primitive predicate > symbol (or operation symbol). Since I don't know a way to develop > mathematics without primitives, I find it quite reasonable - remarkable > and beautiful - that set theory can do the entire job with just one. > > > MoeBlee Well I think that all of that is a dream dear Moe, basing mathematics on only one non defined conept that is e. In reality one cannot understand sets if he/she didn't understand the concept of number first, before we say what is a set? one should solve the question what is number? To me I think that number is a much more basic concept than sets. In reality the concept of number perhaps would be the an indefinable concept as Peano suggested, and basing mathematics on such concept is really rewarding. sets stems from treating Many as one , If you don't understand the meaning of "one" you cannot understand the meaning of set. So what is the meaning of ONE. Zuhair |