The set of All sets
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From: zuhair on 19 Nov 2005 04:20 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. If U can be identical to itself. and so a subset of itself. then P(P(x)) = P(x) so U = { U } P(P(P(x)))= {{U}} This can go forever so ........................P(P(P(x)))= ......................{{{U}}}.................... This means the U is always a proper superset of itself , and equally always a proper subset of itself. I call U a beautiful example of " Unbounded set ". Similar condition happens with P(x) = Singlton x = {x} Now P(P(x))= Singlton {x} = {{x}} which is also a singlton so we will have ....................P(P(P(x)))= ........................{{{x}}}}................... Now if x = nothing or emptyness then ....................P(P(P(x)))= .........................{x}...................= .......{{{{{}}}}........ But isn't that the set of all natural numbers. If that is N , then N cannot catch itself , or N is always a proper subset of itself, and equally N is always a proper superset of itself, so N is an unbounded set. This will have alot of implications. since we define infinite sets according to a certain rule or function , however these functions are True for finite subsets of these infinite sets and since they are true for every finite subset of the infinite set then they are regarded as true of all the infinite set, this all comes from the fact that a set can be a subset of itself but with the above conditions of N here N always do not equal the set union of all its finite sets , it is either a proper subset of them or it is a proper superset to them, but never identical or equal to the set union of all its finite sets. Accordingly the base for reflexion priniciple : that an infinite set can be injected to some proper subset of it would be distroyed. Since no function in current mathematics can catch any infinite set. Zuhair
From: Robert Low on 19 Nov 2005 04:50 zuhair wrote: > Similar condition happens with > > P(x) = Singlton x = {x} Assuming that this made sense, > Now P(P(x))= Singlton {x} = {{x}} which is also a singlton This would still be wrong. P({x})={ {}, {x} } which has two elements. In general, if a set A has n(>=0) elements, P(A) has 2^n elements.
From: boink on 19 Nov 2005 05:17 you're kind of talking gibberish. there is no set of all sets. you could say that there is a _proper class_ of all sets, i.e.: {x: x=x} also, do a google search on Quine atoms.
From: Robert J. Kolker on 19 Nov 2005 06:52 zuhair wrote: > Hello everyone. > > Define U to be the set of all x were x fulfilles the propositional > function > > P(x) = x. You are running on empty. Bob Kolker
From: Robert J. Kolker on 19 Nov 2005 06:53
Robert Low wrote: > zuhair wrote: > >> Similar condition happens with >> >> P(x) = Singlton x = {x} > > > Assuming that this made sense, > >> Now P(P(x))= Singlton {x} = {{x}} which is also a singlton > > > This would still be wrong. P({x})={ {}, {x} } > which has two elements. > > In general, if a set A has n(>=0) elements, P(A) has 2^n > elements. Don't confuse zuhair with facts. Bob Kolker |