The set of All sets
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From: Han de Bruijn on 21 Nov 2005 03:24 zuhair wrote: > Hello everyone. > > 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. Nothing is identical to itself. ????? ???, ????? ????? = Panta rhei, ouden menei (: Heraclitus of Ephesus, 540-480 BC). Everything flows, nothing remains (the same). Han de Bruijn
From: zuhair on 21 Nov 2005 07:02 Han de Bruijn wrote: > zuhair wrote: > > > Hello everyone. > > > > 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. > > Nothing is identical to itself. ????? ???, ????? ????? = Panta rhei, > ouden menei (: Heraclitus of Ephesus, 540-480 BC). Everything flows, > nothing remains (the same). > > Han de Bruijn Ok , even if , then I can define Px <-> x = ~x , ~ means Not. So if U is the set of all the truth values of x in the above function. Then U= The set of all sets ={.......} = { x|Px }, since nothing is identical to itself as you say would imply that everything is identical to what is not itself. Now U itself is not identical to itself as you say. Then U is a truth value of x fulfilling the propositional function Px above Since U is the set of all truth values of x fulfilling Px. Then: U = { U,.....} Now even this {U,...} is not the same as itself as you said Then it should be a member of U so U = { { U,.....},........... } And that can go for ever U = {{ { U,.....},........... },............} Ultimately we will have U= .........{{{{{{U,...},...},....},....},....},....}......... We reached the same result but from a different angle. I am interested in an imaginary subject that is the set of all non-existing sets In reality I believe that U above is the set of all non existing sets, because I believe that everything is identical to itself. I don't know weather that is equivalent to the following U= .......{{{{{ }}}}}......... Since a non existing set is empteness????? But if we say that {} is 0 , { {} } = {1}, { { {} } } = {0,1,} , {{{{}}}}= {0,1,2} Then U = N= { 0,1,2,3,4,5,..........} So N is uncatchable set. One would say that a non existing set is not identical to the empty set But my question is that if X is a "non existing set" Then { X } is an empty set. However this might be wrong. Any comment Zuhair
From: zuhair on 21 Nov 2005 12:36 zuhair wrote: > Han de Bruijn wrote: > > zuhair wrote: > > > > > Hello everyone. > > > > > > 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. > > > > Nothing is identical to itself. ????? ???, ????? ????? = Panta rhei, > > ouden menei (: Heraclitus of Ephesus, 540-480 BC). Everything flows, > > nothing remains (the same). > > > > Han de Bruijn > > Ok , even if , then I can define Px <-> x = ~x , ~ means Not. > > So if U is the set of all the truth values of x in the above function. > > Then U= The set of all sets ={.......} = { x|Px }, since nothing is > identical > > to itself as you say would imply that everything is identical to what > is not itself. > > Now U itself is not identical to itself as you say. > > Then U is a truth value of x fulfilling the propositional function Px > above > > Since U is the set of all truth values of x fulfilling Px. > > Then: U = { U,.....} > > Now even this {U,...} is not the same as itself as you said > > Then it should be a member of U > > so U = { { U,.....},........... } > > And that can go for ever U = {{ { U,.....},........... },............} > > Ultimately we will have > > U= .........{{{{{{U,...},...},....},....},....},....}......... > > We reached the same result but from a different angle. > > I am interested in an imaginary subject that is the set of all > non-existing sets > > In reality I believe that U above is the set of all non existing sets, > because I believe > > that everything is identical to itself. > > I don't know weather that is equivalent to the following > > U= .......{{{{{ }}}}}......... > > Since a non existing set is empteness????? > > But if we say that {} is 0 , { {} } = {1}, { { {} } } = {0,1,} , > {{{{}}}}= {0,1,2} > > Then U = N= { 0,1,2,3,4,5,..........} > > So N is uncatchable set. > > One would say that a non existing set is not identical to the empty > set > > But my question is that if X is a "non existing set" > > Then { X } is an empty set. > > However this might be wrong. > > Any comment > > Zuhair In continuation to that post, I got an interesting idea! The set U above is the set of all non existing set, then is U existing or not? According to ZF set theory the set of all sets do not exist. In reality I think that ZF set theory is concerned with existing sets So in reality what ZF means is The set of all existing sets DO NOT exist. But could it be prooved according to ZF set theory that the set of all non-existing sets DO exist. If so I think this set would be the EMPTY set. So U= { } Now if we define U' as the set of the truth values of x in Px <-> x =x Then U' is the set of all existing sets , and it should not exist, accordingly it is a member of the empty set U. So The power set of the set of all existing sets is the empty set ???? But what is a non-existing set I have the idea that every existing set A has a non-existing set A' Such that if A = { x | Px } Then A' = { x| Px } were A' is defined to contain the false values of x that of coarse do not fulfill Px. Example the set of the first three natural numbers N= {1,2,3} this is an existing set However the set of the first three non -natural numbers N' ={ 1,2,3} this is a non existing set. Do not confuse A' with the complementary set of A. So the set of all non-existing sets is an existing set and it is the NULL SET. Zuhair
From: MoeBlee on 21 Nov 2005 16:59 zuhair wrote: > Han de Bruijn wrote: > > zuhair wrote: > > > > > Hello everyone. > > > > > > 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. > > > > Nothing is identical to itself. ????? ???, ????? ????? = Panta rhei, > > ouden menei (: Heraclitus of Ephesus, 540-480 BC). Everything flows, > > nothing remains (the same). > > > > Han de Bruijn > > Ok , even if , then I can define Px <-> x = ~x , ~ means Not. > > So if U is the set of all the truth values of x in the above function. You mean U is the set of all x such that x not= x. > Then U= The set of all sets ={.......} = { x|Px }, since nothing is > identical > to itself as you say would imply that everything is identical to what > is not itself. If 'x is itself' iff 'x is identical with x', then what you've said doesn't follow. Ax~Ixx does not imply Axy(~Ixy -> Ixy). If nothing were identical with itself, then U would be the set of all sets. > Now U itself is not identical to itself as you say. > Then U is a truth value of x fulfilling the propositional function Px > above U is not a truth value. What you mean is that the truth value is true for U as an argument for the propositional function P. > Since U is the set of all truth values of x fulfilling Px. U is the set of x such that Px. > Then: U = { U,.....} UeU, yes. > Now even this {U,...} is not the same as itself as you said > Then it should be a member of U > so U = { { U,.....},........... } Right. > And that can go for ever U = {{ { U,.....},........... },............} > > Ultimately we will have > > U= .........{{{{{{U,...},...},....},....},....},....}......... No, we'll have U. Your notation above is undefined. We'll have the set of all sets, which has a member itself, which has a member itself, infinitely, if that's what you mean. But U has as members other things besides itself. Also, if the identity axioms don't hold in this situation, then some of the unstated middle steps in these inferences won't be supported, so you won't even get to play around with these things the way you think you can. > We reached the same result but from a different angle. > I am interested in an imaginary subject that is the set of all > non-existing sets > In reality I believe that U above is the set of all non existing sets, > because I believe > that everything is identical to itself. Good. But if everything is identical with itself, then U is the empty set. Calling it the set of 'all non-existing sets' is an unnecessarily bizarre way of expressing it. > I don't know weather that is equivalent to the following > > U= .......{{{{{ }}}}}......... > > Since a non existing set is empteness????? There are descriptions such that no set fits the description. But there is no set that exists but does not exist. If something is a set, then it exists. And your notation: .......{{{{{ }}}}}......... isn't defined > But if we say that {} is 0 , { {} } = {1}, { { {} } } = {0,1,} , > > {{{{}}}}= {0,1,2} > > Then U = N= { 0,1,2,3,4,5,..........} Which U? The set of all everything that is not identical to itself, with the assumption that nothing is identical to itself? Or your undefined notation:.......{{{{{ }}}}}.........? In the first case, U is not {0 1 2...}. In the second case, it's just undefined notation. > So N is uncatchable set. > > One would say that a non existing set is not identical to the empty > set No one I know would. There are no non-existing sets. If something is a set, then it exists, whether it's empty or full to the brim. > But my question is that if X is a "non existing set" > > Then { X } is an empty set. Whatever X stands for, it stands for something that exists. I don't know what system of logic you have in which X can stand for something that does not exist (this is aside from the question of non-referring terms resulting from conditional definitions). > However this might be wrong. It's barely coherent enough to be wrong. > Any comment See above. > Zuhair MoeBlee
From: MoeBlee on 21 Nov 2005 17:26
zuhair wrote: > The set U above is the set of all non existing set, then is U existing > or not? Existing. With the qualification that 'non-existing set' is not in the language of set theories such as Z or ZF. > According to ZF set theory the set of all sets do not exist. > > In reality I think that ZF set theory is concerned with existing sets One would hope so. > So in reality what ZF means is The set of all existing sets DO NOT > exist. ZF doesn't use 'existing' as a predicate. > But could it be prooved according to ZF set theory that the set of all > non-existing sets > DO exist. ZF doesn't have a predicate 'existing'. > If so I think this set would be the EMPTY set. If you change 'non-existing' to 'does not equal itself', then the set of sets that do not equal themselves is the empty set. > So U= { } > > Now if we define U' as the set of the truth values of x in Px <-> x =x You mean U' is the set of x such that x=x. > Then U' is the set of all existing sets , and it should not exist, > accordingly it is a member > of the empty set U. What premise are you using now? That nothing is identical with itself? Or everything is identical with itself? If the former, then we're not in ZF, and I don't know how you'd do all the middle steps with the equality sign. Putting that aside, U' is the univeral set under those conditions. If the later, then, in ZF there is no such set. But what is bizarre is how you've managed to say that non-existent things are members of the empty set. > So The power set of the set of all existing sets is the empty set In ZF? There is no set of all sets to apply the power set operation to. > But what is a non-existing set There is no non-existing set. There are descriptions that no set satisfies. But if somethig is a set, then it exists. > I have the idea that every existing set A has a non-existing set A' I have an idea that every nonset existing a has A A'. > Such that if A = { x | Px } > > Then A' = { x| Px } > > were A' is defined to contain the false values of x that of coarse do > not fulfill Px. No, you just defined A' = A. And sets are not generally false values or true values, unless they are indeed values for a Boolean function. > Example the set of the first three natural numbers N= {1,2,3} this is > an existing set > > However the set of the first three non -natural numbers N' ={ 1,2,3} > this is a non existing > set. That's nonsense that deserves one and only one description: a work of art. > Do not confuse A' with the complementary set of A. They're the same as you defined them. > So the set of all non-existing sets is an existing set and it is the > NULL SET. Bravo! > Zuhair |