The set of All sets
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From: Jonathan Hoyle on 21 Nov 2005 17:33 >> Define U to be the set of all x were x fulfilles the propositional >> function >> >> P(x) = x. >> >> U = Set of All sets , because everything is identical to itself. What logical framework are you using? You can't be using ZFC since the collection of all sets is not a set, and thus not addressable within it. BNG is a conservative extension of ZFC that does allow you to address the collection of all sets, but this collection would be a proper class and not a set. There are some non-Well Founded Set Theories that do give you a set of all sets, but these non-WF sets do not exhibit the same properties as typical ZFC sets. It appears that you have not thought this very well through. You cannot logically deduce anything until you have first decided which logic you wish to perform your deductions upon. Hope that helps, Jonathan
From: zuhair on 22 Nov 2005 02:56 I think I had the mistake of interpretating what is a set? I thought that a set is an equivalence relation between subjects and a predicate all the subjects has, or a one-many equivalence relation between predicate and subjects having that predicate. So if P(x) = y ( this is a subject-predicate function ) x is the subject , y the predicate and P is the rule which identifies the presence of the predicate y in x, to result in the whole statement of " P(x)=y IS True " Now x = P^(-1) y Now P(-1) y is read as the set of y , so set in general is an inverse of a subject - predicate function. P^(-1) y which is the inverse of P(x) , mapps y to All x having y, and of coarse this can be multivalued or single valued or NUL valued. Now if we say their is P' (x)=y , were P' is a rule which identifies the presence of the predicate y' ( y' is a predicate other than y) in x , then P'^(-1) y might be different from the set P^(-1)y , in reality P'^(-1)y = P^(-1) y' so if P ^(-1) y = Z then P^(-1)y' = Z' Of coarse Z' is not necessarily different from Z nor necessarily a complementary set of Z. All the above are examples of true sets or existing sets Now lets have another definition which is in apposit to the above L (x) = y x is the subject , y the predicate and L is the rule which identifies the presence of the predicate ~y in x, to result in the whole statement of " P(x)=y IS false " Now what is L^(-1) y , the result is a False set of y or a non-existant set of y. { L^(-1) y } = { } In reality now I am coming to think that the set of all true sets U is a true set and thus it exists. A non-existent set or false set is something else , and the set of all false sets is a false set also. So { } is the set of any non-existant sets . While U is the set of all true sets. Of coarse the null set of non-existent sets might be U or perhaps any true set???? All of that resulted because I defined set as inverse subject-predicate function , if the function is a truth function then it is a ture set, if it is a falsy function then it makes a false set( or non existent sets ) Of coarse their might be also mixed false and true sets??? Just thoughts Zuhair...............
From: MoeBlee on 22 Nov 2005 14:54 zuhair wrote: > I think I had the mistake of interpretating what is a set? I thought > that a set is an equivalence relation between subjects and a predicate > all the subjects has, or a one-many > equivalence relation between predicate and subjects having that > predicate. > > So if P(x) = y ( this is a subject-predicate function ) > > x is the subject , y the predicate and P is the rule which identifies > the presence > of the predicate y in x, to result in the whole statement of " P(x)=y > IS True " > > Now x = P^(-1) y Now P(-1) y is read as the set of y , so set in > general is an inverse of a subject - predicate function. > > P^(-1) y which is the inverse of P(x) , mapps y to All x having y, and > of coarse this can be multivalued or single valued or NUL valued. > > Now if we say their is P' (x)=y , were P' is a rule which identifies > the presence of the > predicate y' ( y' is a predicate other than y) in x , then P'^(-1) y > might be different from the set P^(-1)y , > > in reality P'^(-1)y = P^(-1) y' > > so if P ^(-1) y = Z then P^(-1)y' = Z' > > Of coarse Z' is not necessarily different from Z nor necessarily a > complementary set of Z. > > All the above are examples of true sets or existing sets > > Now lets have another definition which is in apposit to the above > > > L (x) = y > > x is the subject , y the predicate and L is the rule which identifies > the presence > of the predicate ~y in x, to result in the whole statement of " P(x)=y > IS false " > > Now what is L^(-1) y , the result is a False set of y or a > non-existant set of y. > > { L^(-1) y } = { } > > In reality now I am coming to think that the set of all true sets U is > a true set and thus it exists. > > A non-existent set or false set is something else , and the set of all > false sets is a false set also. > > So { } is the set of any non-existant sets . > > While U is the set of all true sets. > > Of coarse the null set of non-existent sets might be U or perhaps any > true set???? > > All of that resulted because I defined set as inverse subject-predicate > function , if the function is a truth function then it is a ture set, > if it is a falsy function then it makes > a false set( or non existent sets ) > > Of coarse their might be also mixed false and true sets??? > > Just thoughts > > Zuhair............... I get it now. Of coarse, it's a hoax. No one, of coarse, could possibly be so outlandishly discombobulated. Of coarse, of coarse, what else could it be? Of coarse it is. MoeBlee
From: zuhair on 27 Nov 2005 04:59 MoeBlee wrote: > zuhair wrote: > > I think I had the mistake of interpretating what is a set? I thought > > that a set is an equivalence relation between subjects and a predicate > > all the subjects has, or a one-many > > equivalence relation between predicate and subjects having that > > predicate. > > > > So if P(x) = y ( this is a subject-predicate function ) > > > > x is the subject , y the predicate and P is the rule which identifies > > the presence > > of the predicate y in x, to result in the whole statement of " P(x)=y > > IS True " > > > > Now x = P^(-1) y Now P(-1) y is read as the set of y , so set in > > general is an inverse of a subject - predicate function. > > > > P^(-1) y which is the inverse of P(x) , mapps y to All x having y, and > > of coarse this can be multivalued or single valued or NUL valued. > > > > Now if we say their is P' (x)=y , were P' is a rule which identifies > > the presence of the > > predicate y' ( y' is a predicate other than y) in x , then P'^(-1) y > > might be different from the set P^(-1)y , > > > > in reality P'^(-1)y = P^(-1) y' > > > > so if P ^(-1) y = Z then P^(-1)y' = Z' > > > > Of coarse Z' is not necessarily different from Z nor necessarily a > > complementary set of Z. > > > > All the above are examples of true sets or existing sets > > > > Now lets have another definition which is in apposit to the above > > > > > > L (x) = y > > > > x is the subject , y the predicate and L is the rule which identifies > > the presence > > of the predicate ~y in x, to result in the whole statement of " P(x)=y > > IS false " > > > > Now what is L^(-1) y , the result is a False set of y or a > > non-existant set of y. > > > > { L^(-1) y } = { } > > > > In reality now I am coming to think that the set of all true sets U is > > a true set and thus it exists. > > > > A non-existent set or false set is something else , and the set of all > > false sets is a false set also. > > > > So { } is the set of any non-existant sets . > > > > While U is the set of all true sets. > > > > Of coarse the null set of non-existent sets might be U or perhaps any > > true set???? > > > > All of that resulted because I defined set as inverse subject-predicate > > function , if the function is a truth function then it is a ture set, > > if it is a falsy function then it makes > > a false set( or non existent sets ) > > > > Of coarse their might be also mixed false and true sets??? > > > > Just thoughts > > > > Zuhair............... > > I get it now. Of coarse, it's a hoax. No one, of coarse, could possibly > be so outlandishly discombobulated. Of coarse, of coarse, what else > could it be? Of coarse it is. > > MoeBlee Is their anything called " a group of sets should be contained by a set " I mean is that consistent with ZF set Theory. Zuhair
From: Jonathan Hoyle on 27 Nov 2005 21:33
In ZF Set Theory with no urelements, sets contain only other sets and nothing else. The set of natural numbers, or example, is just a set of sets. If you mean *all sets* then ZF and ZFC do not allow for the set of all sets. This must be the case, since if the collection of all sets were a set, it must therefore contain itself as a member, which violates the Axiom of Foundation (AF), one of the axioms of ZF. In a conservative extension of ZFC called NBG (vonNeumann-Bernays-Godel), you can have a collection of all sets, but this collection is not a set, but rather a proper class, for the reasons stated above. (All sets are classes, and those classes which are not sets are called proper classes). NBG is equi-consistent to ZFC; that is to say, if NBG is inconsistent, then it is only because ZFC is as well. To have "the set of all sets", you must work within a set theory which does not contain AF. These are called "non-Well Founded" Set Theories, and sets which violates AF are called "non-well founded sets". Obviously non-well founded sets in these theories do not exist in ZF, and thus you cannot assume traditional ZF properties to them. Hope that helps, Jonathan Hoyle |