The set of All sets
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From: zuhair on 11 Dec 2005 00:48 Lee Rudolph wrote: > "zuhair" <zaljohar(a)yahoo.com> writes: > > >Dear Jonathan Hoyle: > > > >In a previous message you said that a set > >is a collection which follows ZFC axioms. > > > >But ZFC axioms speakes of sets. > > > >For example: axiom of extensionality > >states that:-Two sets are the same if and only if they have the same > >elements. > > > > > >Don't you think that there is something cyclical > >in what you are saying. > > Accepting your implicit statement as true for the sake of > argument--so what? Not all circles are vicious. > > Lee Rudolph But this one is vicious. Zuhair
From: Jonathan Hoyle on 12 Dec 2005 00:31 >> In a previous message you said that a set >> is a collection which follows ZFC axioms. >> >> But ZFC axioms speakes of sets. >> >> For example: axiom of extensionality >> states that:-Two sets are the same if and >> only if they have the same elements. >> >> Don't you think that there is something >> cyclical in what you are saying. Fair enough, I was speaking rather informally. A little more introspectively, ZFC is a theory describing collections which it calls sets. (Of course there are other theories which may call different collections as sets.) Only ZFC-style sets are ever discussed within the theory. If you wish a formal definition, then it will describe ZFC as a set of rules involving string manipulations, given one set of well-defined symbols, you can construct another. That it can be associated with what we mathematically refer to as "sets" is completely independent and can be viewed as nothing but coincidence.
From: zuhair on 12 Dec 2005 05:52 Jonathan Hoyle wrote: > >> In a previous message you said that a set > >> is a collection which follows ZFC axioms. > >> > >> But ZFC axioms speakes of sets. > >> > >> For example: axiom of extensionality > >> states that:-Two sets are the same if and > >> only if they have the same elements. > >> > >> Don't you think that there is something > >> cyclical in what you are saying. > > Fair enough, I was speaking rather informally. A little more > introspectively, ZFC is a theory describing collections which it calls > sets. (Of course there are other theories which may call different > collections as sets.) Only ZFC-style sets are ever discussed within > the theory. > > If you wish a formal definition, then it will describe ZFC as a set of > rules involving string manipulations, given one set of well-defined > symbols, you can construct another. That it can be associated with > what we mathematically refer to as "sets" is completely independent and > can be viewed as nothing but coincidence. You are only confusing me. In a previous message you said that "set" is a collection which satisfies ZFC axioms. Now All ZFC axioms contains a term called "set" , so in order to understand these axioms we should know beforhand what do the term " set " mean, but as you said this term is only defined as a collection which satisfies ZFC axioms, clearly cyclical approach. Now though I didn't understand what you are saying , but it seems as if you want to say that the term "set" writtin in the ZFC axioms is not the same as "set" defined by these axioms. And you use the word " string manipulations" to explain the word set present in ZFC axioms. I demand you should clarify your reply. Zuhair
From: MoeBlee on 12 Dec 2005 15:36 zuhair wrote: >> Now All ZFC axioms contains a term called "set" No axioms of ZFC mention 'set'. One can define the predicate 'is a set' or even omit having such a predicate. Nothing in plain ZFC depends on having a predicate 'is a set'. One should not confuse informal and intuitive mention of 'sets' with 'is a set' as a formal predicate of the langugae. The only primitives of the language of ZFC are the 2-place predicates 'equal to' and 'is a member of'. (We can even define 'equal to' and have only 'is a member of' as the only primitive if we like.) Anything else can be definied precisely and without circularity. MoeBlee.
From: Herman Rubin on 12 Dec 2005 20:20
In article <1134419772.606126.252290(a)g49g2000cwa.googlegroups.com>, MoeBlee <jazzmobe(a)hotmail.com> wrote: >zuhair wrote: >>> Now All ZFC axioms contains a term called "set" >No axioms of ZFC mention 'set'. One can define the predicate 'is a set' >or even omit having such a predicate. Nothing in plain ZFC depends on >having a predicate 'is a set'. One should not confuse informal and >intuitive mention of 'sets' with 'is a set' as a formal predicate of >the langugae. The only primitives of the language of ZFC are the >2-place predicates 'equal to' and 'is a member of'. (We can even define >'equal to' and have only 'is a member of' as the only primitive if we >like.) Anything else can be definied precisely and without circularity. >MoeBlee. In ZF, everything is a set. In ZFU, there are individuals also, so everything is either a set or an individual. In NBG, everything is a class, and a set is a class which is an element of a class. Similar remarks to the above apply to NBGU. The axiom of choice has no effect in either. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Department of Statistics, Purdue University hrubin(a)stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 |