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From: MoeBlee on 12 Dec 2005 23:05 Herman Rubin wrote: > 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. I agree with all of that and nothing that I posted contradicts it. (I say that only because I don't know whether you intended for your post to be an addition to what I said or a correction of it.) Thanks, MoeBlee
From: zuhair on 13 Dec 2005 05:02 MoeBlee wrote: > Herman Rubin wrote: > > 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. > > I agree with all of that and nothing that I posted contradicts it. (I > say that only because I don't know whether you intended for your post > to be an addition to what I said or a correction of it.) > > Thanks, > > MoeBlee hmmm.............., let's see: axiom of extensionality states that:-Two "SETS" are the same if and only if they have the same elements. What is the meaning of SETS here. About predicate logic you are using , non of ZFC axioms mentions that explicitely. Zuhair
From: Ross A. Finlayson on 13 Dec 2005 06:16 zuhair wrote: > MoeBlee wrote: > > Herman Rubin wrote: > > > 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. > > > > I agree with all of that and nothing that I posted contradicts it. (I > > say that only because I don't know whether you intended for your post > > to be an addition to what I said or a correction of it.) > > > > Thanks, > > > > MoeBlee > > hmmm.............., let's see: > > axiom of extensionality > states that:-Two "SETS" are the same if and only if they have the same > elements. > > What is the meaning of SETS here. > > About predicate logic you are using , non of ZFC axioms mentions that > explicitely. > > Zuhair Hi Zuhair, "MoeBlee", The predicate logic is basically beneath ZFC. ZFC has what are called axioms, the axioms of ZFC. Those are also called "non-logical" or "proper" axioms. The "logical" "axioms" are combinations of true and false via union and intersection and their results as true or false, i.e. the truth tables. T and T => T T and F => F F and T => F F and F => T T or T => T T or F => T F or T => T F or F => T T nor T => F T nor F => F F nor T => F F nor F => F T xor T => F T xor F => T F xor T => T F xor F => F or ~ = and or nor xor T ~ T T T F F T ~ F F T F T F ~ T F T F T F ~ F T T F F Actually those aren't the logical axioms per se, they're just not very different from them. Also, they don't have the same definition as the standard ones, where for example the column for "and" would be TFFF and for "or" TTTF, and they generally have a different representation and the above might just be seen as wrong. In electronic digital logic the terms to consider are AND, OR, NAND, and NOR, and XOR and XNOR. They're all a result, or derivative from, tautology, where tautology is an equality or identity, and "not". So then the "logical" axioms are basically not under discussion in most considerations of the "non-logical" axioms. They're basically assumed to hold because of the "law of the excluded middle", LEM or the LEM, that something is either true, or false, and not both, and not neither. Some statement of a subject x and predicate P or P(x) is either true, or false, there is no middle ground, the middle is excluded. That's the law of excluded middle, for if something were both true and false it would confuse the truth tables. Then, there are various considerations of indeterminate values that are not necessarily true nor false, some call such a notion U for Undeterminate. In the West, truth tables are normally blamed on De Morgan, and sometimes Boole or Venn. Those are important, key, and critical, and used implicitly in all decision-making. Then the undeterminate values are a fringe thing that started in some formality with Kleene and Lucasiewicz. http://en.wikipedia.org/wiki/Truth_table http://www.google.com/search?q=%22logical+axiom%22 There's some good information in that, and almost undoubtedly some bad. Ross
From: Dave Seaman on 13 Dec 2005 08:16 On 13 Dec 2005 02:02:23 -0800, zuhair wrote: > MoeBlee wrote: >> > 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. >> > > hmmm.............., let's see: > axiom of extensionality > states that:-Two "SETS" are the same if and only if they have the same > elements. No. You can find the axioms of ZFC at <http://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html>. The axiom of extensionality is the first one listed. Where do you see the word "set"? -- Dave Seaman U.S. Court of Appeals to review three issues concerning case of Mumia Abu-Jamal. <http://www.mumia2000.org/>
From: MoeBlee on 13 Dec 2005 11:42
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 |